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If n ■Hiimiimm I
' c,'*^' I 3 '3 2044"096990~536
- ii ' -
I ^- ^
c
HARVARD COLLEGE
LIBRARY
THE ESSEX INSTITUTE
TEXT-BOOK COLLECTION
GIFT OF i^- ~
GEORGE ARTHUR PUMPTON J^'"'' ^f/^.y
OF NEW YORK
JANUASr 2S. I92«
r
J\irv /6. I^-^O . , ^ %.,n U Mela . liM^ ^
ELEMENTARY TREATISE
OB
ARITHMETIC,
TAZSH
PRINCIPALLY FROM THE. ARITHMETIC
OF
S. F. LACROIX>
AND
TAAJTBLATED FROM THR FRRSCH WITH 8VCH ALTKRATIOVB AVI>
ADDITIONB AB WRRR VOVSD HXCR8SARY IV ORDER TO ADAPT
XT TO THE VBR OS AMRRICAJT BTDDRSTS.
BY JOHN FARRAR,
Mfttunl Plulotophf in the UniTenity «t Cainbcidgie*
TBZRD BDZTXOlfy
CORRSGTBD AND BOMSWHAT RNLAROSD.
CAKBRIDGE, N. E.
.* f ... J
ALF, AT TBI VinYSRSITT PRESS.
mihj CmiBuags, BlUiud, lie Co.
8tN(9li Bottoiii
1825.
''I
i
/
E-<i<*ji,T H y. ^ r". Tii
DISTRICT OF MASSACHUSETTS, TO WIT :
Dittrict CUrk'a Office.
Bb it KEMBirBBiiED, that on the twenty-first day of June, A. D. 1826, in the
forty-ninth year of the Independence of the United States of America, Cumminni
Hiltiard, k. Co. of the said district, have deposited in this office the title of a book|
the right whereof they claim as proprietors, in the words following, to wit :
" An elementary treatise on Arithmetic, taken principally from the Arithmetic
of S. F. Lacroiz, and translated from the French, with sUch Alterations and Ad-
ditions as were found necessary in order to adapt it to the Use of American
Students. By John Farrar, Pn>fesr>r of Matfaematicft and Natural Philosophy in
the University at Cambridge. Third edition^ corrected and iomewfaat enlarged/'
In conformity to the Act of the Congress of the United States, entitled,-^' An
Act for the encouragement of learning, by securing the copies of maps, charts, and
books, to the authors and proprietors of such copies, during the times therein
mentioned :" and also to an Act, entitled, " An Act, supplementary to an Act*
entitled, * An Act for the edcouragement of learning, by securing the copies of
maps, charts, and books-, to the authors And proprietors of such copies, during
the times therein mentioned ;' and eitending the benefits thereof to the arts
of designing, engraving, and etchiiig hblorical and other prints.**
JNO. W. DAVIS,
Clerk of the DiUriet of Matfo/chuadU.
ADVERTISEMENT.
The first priociples, as well as the more difficult parts of Mathe-
maticSf have, it is thought, been more fully and clearly eiplaiued by
the French elementary writers, than by the English ; and among
these, Lacroix has held a very distinguished place. His treatises
have been considered as the most complete, and the best suited to
those who are destined for a public education. They have receiv*
ed the sanction of the government, and have been adopted in the
principal schools, of France. The following translation is from the
thirteenth Paris edition. The original being written with reference
to tlie new system of weights and measures, in which the different
denominations proceed in a decimal ratio, it was found necessary to
make considerable alterations and additions, to adapt it to the meas-
ores in use in the United States. The several articles relating to
the reduction, addition, subtraction, multiplication, and division of
compound numbers, have been written anew ; a change has been
made in many of the examples and questions, and new ones hav^
been introduced aAer most of the rules, as an exercise for the
learner.
Cambridge^ Aug. 1818.
CONTENTS-
Of Numeration.
General remarks on the different kinds of magnitade or quantity 1
Of nomber --...-•.. 1
Of spoken numeration ...... . 2
Manner of representing numbers by figures^ or written numera-
tion - - -- - - -4
Of readii^ numbers - - - - - - 6
Of abstract and concrete numbers - - - - 8
Of Addition.
Of the principles on which addition is founded
Cfreneral rule for performing addition
8
- 10
Of Subtraction.
m
Of the principles on which it is founded - - - 1 1
Explanation of the terms, remainder^ excess, and difference - 11
General rule for subtraction - - - - - 14
Method of proving addition and subtraction - - - 14
Of Multiplication.
The origin of multiplication - - - -: 16
An explanation of the terms muUiplicandy muUiplierj product,
eskdfaciors ... . . . 16
Of the principles on which multiplication is performed - 17
The table of Pythagoras, containing the products of any two fig-
ures - - - - ..-17
Formation of this table . - - - - - 17
Remarks, from which it is inferred, that a change in the order of
the factors does not affect the product - - - 18
ti CanienU.
Rale for multipljing a nambek*, consisting of seyeral figures, by a
single figure - - - - - - 20
To maltiply by 10, 100, 1000, &c - - - - 21
Rule for multiplying by a number consistiiig of a single digit and
any number of ciphers ... - 82
General rule for multiplication - - - - 23
Manner of abridging the process, when both factors are terminat-
ed by ciphers ... - - 23
Of Division.
The origin of division - - - - - - 24
Explanation of the terms, dividend j cKvtfor, and ^tiemt » 26
Of the principles on which division is founded - - 25
Mode of proceeding, when the divisor consists of sereral figures 29
Geueral rule for division ... . • -30
Method of abbreviating the process of division - - - 31
When both the divisor and dividend have ciphers on the right 31
Multiplication and division mutually prove each other - 32
Of Fractions.
The origin of fractions .... - 32
The manner of reading and writing fractions - . - - 34
An explanation of the terms, numerator and denominator 34
Of the changes which a fraction undergoes, by the increase or
diminution of one of its terms - - - - 35
A table representing the changes which take place in a fraction,
by the multiplication or division of either of its terms - 36
The value of a fraction not altered by multiplying or dividing
both its terms by the same number • - - 36
To simplify a fraction without altering its value - - 37
The g^atest common divisor of two numbers ... 38
General mle for finding the greatest common divisor - - 40
To distinguish the numbers divisible by 2, 6, or 3 . - - 41
Of prime numbers - - - - - 42
General signification of the term muhiplicaiion - - 44
To multiply a whole number by a fraction - - - - 44
To find the whole number contained in a fraction - - 45
To reduce a whole namber to a fraction - - - 46
ConUnif. vii
To multiply one fraction by another • ... 46
Of compound fractions • .... 47
Of division in general . . - • . 47
Of the division of a whole nund>er by a fraction - • 48
fo divide one fraction by another ... . 48
Of the addition and subtraction of fractions • - - dO
To reduce fractions to a common denominator * 50
Of the addition and subtraction of mixed numbera - - 61
The product of several factors' not changedi by changing the or-
der in which they are multiplied - - - 5S
•
Of Decimal Fractions.
The origin of decimal fractions - - - - 54
The manner of reading and writing decimals * - - 56
A number containing decimals, not altered by annexing ciphers 56
Addition of decimals • - - - - 56
Subtraction of decimals - - - - - 57
The effect of changing the place of the decimal point - • 58
To multiply a number containing decimals, by a whole number 60
The multiplication of one decimal by another • • - 60
To divide a decimal number by a whole number - - - 61
To divide one decimal by imother - - - • 62
Method of approximating the quotient of a division by decimals « 62
Able. — Method of finding the value of the quotient of a diviidon
in fractions of a given denomination • ... 63
To reduce vulgar fractions to decimals .... 63
Aou. — On the changing of one fraction to another of a lower
denomination - - - - - - 64
Of periodical decimak - - * • - - 65
Tables of Coin^ Wetgkt^ and Measure.
Of Federal Money - • - • . - M
English Money - - - - - - 69
Troy Weight - - - - - - 69
Apothecaries' Weight - - - - - 69
Avoirdupois Weight - - - - - 70
Long Measure - - - ---71
Cloth Measure - - - - - - 71
' >,»^'" j 3 3^044 096 990 536
HARVARD COLLEGE
LIBRARY
THE ESSEX INSTITUTE
TEXT-BOOK COLLECTION
I:
GIFT OF
GEORGE ARTHUR PUMPTON
OF NEW rOKK
MNDARY 2S, 1924
\>f
J • /'^ • ♦//'''-'••^' /^'^'''^''A *^*'^^^fit^.
ELEMENTARY TREATISE
o«
ARITHMETIC,
TAZSH
PRINCIPALLT FROM THE. ARITHMETIC
OF
S. F. LACROIX>
▲VD
TRAVBLATBD FROM THE VRXITCH WITH 8VCH ALTERATIOVB AVI>
ADDITI0S8 A8 WERE VOVSO EECBWARY IV ORDER TO ADAPT
XT TO THE VaE OE AMERICAE STUDESTS.
BY JOHN FARRAR,
FraftMor iif MMhemttiet and Matunl Pldlaiopliy in the Unirenity at Cainbcidgie*
TBZRD BDZTZOlfy
CORRSGTXD AND SOMEWHAT KRI.AEOED.
CAMBRIDGE, N. E.
rmiTTED BT BILLIARD AND METCALF, AT THE ViaVERSITT PRESS.
Sold by W. Hilliard, Cmnbridce, and by Cummuigt, HiUiard, 1^ Cob
No. 134 Wjttblastoa StrecH, BoMon.
/
1825.
fl
X Contents,
Mhuner o{ wriiing nnmhers in equidifferencit - «- •US
Questions for practice .. . . • » 118
Of Alligation.
The principles of medial aUigation explained - - 120
Illustrated by examples - - - . . 120
Alligation alternate explained - - - - 121
Examples for illustration - - - - - 122
Mucellaneow Q^uestions. - - 124
M'oie, — Summation of a Continued Fraction - - - 129
Appendix.
Tables of various weights and measures - • - 133
New French weights and measures - - - 133
Reasons for adopting the decimal gradation - - - 133
The measures of length - - - . ^133
The measures of capacity - - - - - 134
Weights -w - - - - -134
Land measure - - • - - -135
The division of the circle - - - - - 136
The decimal system of coin - - - - 135
Divisions of time - - - - - 135
Scripture long measure - - - • - 139
Grecian long measure reduced to English - - 136
Jewish long or itinerary measure - - - - 137
Roman long measure reduced to English - • - ] 37
Attic dry measure reduced to English - - - 139
Attic measures of capacity for liquids, reduced to English wine
measure ...... 138
Measures of capacity for liquids, reduced to English wine mea-
sure ... - . ---138
Jewish dry measure reduced to English - - - - 139
Jewish measure of capacity for liquids, reduced to English wine
measure -...-- 139
Ancient Roman land measure - - - • 139
Roman dry measure reduced to English - - - 139
Table of the principal gold and silver coins, containing their
weights, fineness, pore contents, current value, &c. - 140
Explanation of the Roman Numerals.
One
Two
Three
Four
FiTC
Six
Seyea
Eight
Vim
Ten
Twenty
Thirty
Forty
Fifty
Sixty
Seventy
Eighty
Ninety
Hundred
Two hundred
Three hundred
Four hundred
n»
m
m
V
VI}
vii
viu
IX
X
XX
XXX
XL
L
LX
LXX
LXXX
xc
c
cc
ccc
cccc
* As often as any eharacter is repeated, so majiy times its value is re-
peated.
t A less character before a greater diminishes its yalae.
t A less eharacter after a greater increases its value. ■
i
zii
Rotnan NvmeraU.
Fiye handred
Six hundred
Seven hundred
Eight hundred
Nine hundred
Thousand
Eleven hundred
Twelve hundred
Thirteen hundred
Fourteen hundred
Fifteen hundred
Two thousand
Five thousand
Six thousand
Ten thousand
Fifty thousand
Sixty thousand
Hundred thousand
Million
Two millions
&c. &c.
D or DI»
DC
DCC
DCCC
DCCCC
MorClOt
MC
MCC
MCCC
MCCCC
MD
MM
100 : or Tj
VI
XorCCIOO
1000
LX
C or CCCIOOO
M or CCCCIOOOD
MM
* For every affixed this becomes ten times as many.
t For every C and D put one at each end) it is increased ten times.
i A line over any number increases it 1000 fold.
ELEMENTARY TREATISE
ON
ARITHMETIC.
Numeration.
1. i¥ COMPARISON of the different objects, that come within the
reach of our senses, soon leads us to perceive, that, in all these
objects, there is an attribute, or quality, by which they can be
supposed susceptible of increase or diminution ; this attribute is
magnitude. It generally appears in two different forms. Some-
times as a collection of several similar things, or separate parts,
and is then designated by the word number.
Sometimes it presents itself as a whole, without distinction of
parts; it is thus, that we consider the distance between two
points, or the length of a line extending from one to the other,
as also the outlines and surfaces of bodies, which determine their
figure and extent^ and finally this extent itself.
The proper characteristic of this last kind of magnitude, is
the connexion or union of the parts, or their continuity ; whilst in
number we consider how many parts there are ; a circumstance
to which the word quantity at first had relation, though after*
wards it was applied to magnitude in general, magnitude con-
sidered as a whole being called continued quantity^ to distinguish
it from number, which is called discrete, or discontinued quantity.
3. All that relates to magnitude is the object of mathematics ;
numbers, in particular, are the object of arithmetic.
Arith. I
2 Arithmetic.
Continued magnitude belongs to geometry^ which treats of the
properties presented by the forms of bodies, considered with
regard to their extent*
3. Number, being a collection of many similar things, or many
distinct parts, supposes the existence of one of these things, or
parts, taken as a term of comparison, and this i& called unity*
The most natural mode of forming numbers is, to begin with
joining one unity to another, then to this sum another ; and con-
tinuing in this manner, we obtain collections of units, which are
expressed by particular names ; these names taken together,
which vary in different languages, compose the spoken nuniera-
Hon*
4. As there are no limits to the extension of numbers, since
however great a number may be, it is always possible to add an
unit to it, we may easily conceive that there is an infinity of
different numbers, and consequently, that it would be impossible
to express them, in any language whatever, by names that should
be independent of each other.
Hence have arisen nomenclatures, in which the object has
been, by the combinations of a small number of words, subject
to regular forms, and therefore easily remembered, to give a
great number of distinct expressions.
Those, which are in use in the [English language,] with few
exceptions, are derived from the names assigned to the nine first
numbers and those afterwards given to the collections of ten, a
hundred^ and a thousand units.
The units are expressed by
wi€, (too, three^ four^ five, six, seoen, eight, nitie^
The collections of ten units, or tens, by
ten, twenty, thirty , forty, fifty, sixty, seventy, eighty, ninety^
The collections of ten tens, or hundreds, are expressed by
names borrowed from the units ; thus wc say,
hundred, two hundred, thru hundred, .... nine hundred.
The collections of ten hundreds, or thousands, receive their
Tfwntration^ 3
•denominations from the nine first numbers and from the collec-
tions of tens and hundreds ; thus we say,
thousand^ tv)o thousand •..••• nine thousand^
ten thousand^ twenty thousand^ ($rc.
hundred thousand^ two hundred thousarid^ ^c.
The collections of ten hundred thousands, or of thousands of
thousands, take the name of millions^ and are distinguished like
the collections of thousands.
The collections of ten hundreds of millions, or of thousands of
millions, are called billions^ and are distinguished, like the collec-
tions of millions^
t The idea of number is the latest and most difficult to form. Be-
fore the mind can arrive at such an abstract conception, it must be
familiar with that process of classification, bj which we successively
remount from individuals to species, from species to genera, and from
genera to orders. The savage is lost in his attempts at numeration,
and significantly expresses his inability to proceed by holding up his
expanded fingers, or pointing to the hairs of his head.
Nature has furnished the great and universal standard for compu-
tation in the fingers of the hand. All nations have accordingly
reckoned by fives ; and some barbarous tribes have scarcely advanc-
ed any further. Afler the fingers of one hand had been counted once,
it was a second and perhaps a distant step to proceed to those of the
other. The primitive words, expressing numbers, did not probably
exceed five. To denote nx, seven^ ^ight^ ^ud nine^ the North Ameri-
can Indians repeat the ^ye with the successive addition of one, two,
three, and four ; could we safely trace the descent and affinity of the
abbreviated terms denoting the numbers from five to ten, it seems
highly probable, that we should discover a similar process to have
taken place in the formation of the most refined languages.
The ten digits of both hands being reckoned up, it then became
necessary to repeat the operation. Such is the foundation of our
decimal scale of arithmetic. Language still betrays by its structure
the original mode of proceeding. To express the numbers beyond
ten, the Laplanders combine an ordinal with a cardinal digit Thus,
eleven, twelve, &c. they denominate second ten and one, second ten
and two, &c. and in like manner they call twenty-one, twenty-two,
&c. third ten and one, third ten and two, &c. Our term eUven 'v^
4 Arithmetic.
Each of the names just mentioned is considered as forming an
unit of an order more elevated according as it is removed from
the place of simple units. The names ten and hundred are continu-
ally repeated, and we have no occasion for new names, such as
thousand^ million, billion, except at every fourth order. The same
law being observed, to billions succeed trillions, quadrillionsj
quintillions, &,c. each, like billions, having its tens and hundreds.
Numbers expressed in this manner, when more than one word
enters into the enunciation of them, are separated into their
respective orders of units, mentioned above ; for instance, the
number expressed hjfive hundred thousand three hundred and two,
is separated into three parts, viz. jive hundreds of thousands, three
hundreds of simple units, and ttoo of these units*
5. The length of the expression, written in words, when the
numbers were large, occasioned the invention of characters, ex-
clusively adapted to a shorter representation, and hence origi-
nated the art of expressing numbers in writing by these charac-
ters, called figures, or written numeration.
The laws of the written numeration, now used, are very anal-
ogous to those of the spoken numeration. In it the nine first
numbers arc each represented by a particular character, viz.
12 3 4 56 7 89
one, two, three, four, five, six, seven, eight, nine.
supposed to be derived from ein or one, and liben, to remain, and
to signify one, leave or set aside ten, T\velve 4l of the like deriva-
tion, and means two, laying aside the ten. The same idea is suggest-
ed by our termination 1y in the words, twenty, thirty, &c. This syl-
lable, altogether distinct from ten, is derived from ziehen, to draw,
and the meaning of twenty is, strictly speaking, two drawings, that iS}
the hands have been twice closed and the fingers counted over.
* After ten was firmly established, as the standard of numeration, it
seemed the most easy and consistent to proceed by the same repeaL
ed composition. Both hands being closed ten times would carry the
reckoning up to a hundred. This word, originally hund, is of uncer-
tain derivation ; but the term, thousands which occurs at the next
stage of the progress, or the hundred added ten times, is clearly
traced out, being only a contraction of duis hund, or twice hundred,
that is, the repetition or collection of hundreds. See Edinburgh Re-
view, vol. xvni. art. vii.
J^umeration. 5
When a number consists of tens and units, the characters repre-
senting the number of each are written in order from left to
right, beginning with the tens. The number forty-seven, for
instance, is written 47 ; the first figure on the left, 4, denotes the
four tens, and consequently a value ten times greater than it
would have standing alone ; while the figure 7, placed on the
right, expressing seven units, possesses only its original value.
In the number thirty-three, which is written 33, we see the
figure 3 repeated, but each time tvith a diflferent value ; the value
of the 3 on the left is ten times greater than the value of that on
the right.
This is the fundamental law of our written numeration, that
a removal^ of one place^ towards the left increases the value of a
figure ten times.
If it were required to express fifty, or five tens, as there are
no units in this number, there would be nothing to write but the
figure 5, and consequently it would be necessary to show, by
some particular mark, that in the expression of this number, the
figure ought to occupy the first place on the left. To do this we
place on the right the character 0, cipher^ or nought, which of
itself has no value, and serves only to fill the place of the units,
which are wanting in the enunciation of the proposed number.
6.; Thus with ten characters, by means of the rule before laid
down concerning the value which figures assume, according to
the places they occupy, we can express all possible numbers^
With two figures only, we can write all, as far as to nine tens
and nine units, making 99, or ninety-nine. After this comes the
hundred, which is expressed by the figure 1, put one place far^
iher towards the left, than it would be, if used to express tens
only ; and to denote this place, two ciphers are placed on the •.
right, making 100.
The units and tens, afterwards added to form numbers greater
than 100, take their proper places; thus a hundred and one will
be written in figures 101 ; a hundred and eleven, 111. Here the
same figure is three times repeated, and with a difierent value
each time ; in the first place on the right it expresses an unit,
in the second, a ten, in the third, a hundred. It is the same
6 Ariihmeticw
with the number 22% 333, 444, Sec. Thus, in consequence of
the rule laid down before when speaking of units and tens, the
same figure expresses units ten limes greater,, in proportion as it is
removed from right to left^ and^ hy a simple change ofplace^ acquires
the power of representing^ successively^ all the different collections of
units, which can enter into the expression of a number*
7. A number dictated, or enunciated, h written, then, by plac-
ing one after the other, beginniig at the left, the figures which
express the number of units of each collection ; but it is neces-
sary to keep in mind the order in which the collections succeed
each other, that no one may be omitted, and to put ciphers in the
room of those which are wanting in the enunciation of the num-
ber to be written. If, for example, the number were three hun-
dred and twenty four thousand, nine hundred and four, we should
put 3 for the hundreds of thousands, 2 for the twenty thousands,
or the two tens of thousands, 4 for the thousands, 9 for the hun-
dreds ; and as the tens come immediately after the hundreds,
and are wanting in the given number, we should put a cipher in
the room of them, and then write the figure 4 for the units ; we
should thus have 324904.
In the same way, writing ciphers in the place of tens of thou-
sands, thousands, and tens, which arc wanting in the number five
hundred thousand three hundred and two, we should have
500302.
8. When a number is written in figures, in enunciating it, or
expressing it in language, it is necessary to substitute for each
of the figures the word which it represents, and then to mention
the collection of units, to which it belongs according to the place
it occupies. The following example will illustrate this ;
•'^•I'j^'?.
J^Cumeratian* 7
TrilKoBs. Billions.t Millions. Thousands. Units.
2 4, 8 9 7, 3 2 1, 6 8 0, 3 4 6
HHfflHWffiHSffiHH-aHc!
c/i
oa
a O a»
CO B »
a.
en
CO
The figures of this number are divided by commas, into por-
tions of three figures each, beginning at the right ; but the last
division on the left^ which in the present instance has but two
figures, may sometimes have but one. Each of these divisions
corresponds to the collections designated by the words unitj thou^
sanAf million, btV/ion, trillion, and their figures express succes-
sively the units, tens, and hundreds of each. Thtis, the expression
t The folio wing" are the denominations generally adopted in Eng-
lish and American b«okd on arithmetic :
A
A.
A.
^
r"
'N
/"
"\
2
4
5,
9
2
4,
8
9
7,
3
2
1,
5
8
0,
3
4
6
X
H
H
«
•^
CO
a
H
H
ffi
•^
S
W
H
H
W
H
MM
a
o
tr
c
A
V*
c
A
p*
c
(t
^«
(»
p-
p
(^
P
IHID
a
S
o
D
D
^iri*
p
S
o
p
p
^va
£3
p
o
p
p
^
1
o
c
&<
CD
o
O'
CD
c
&
CO
CL
CD
p
Cb
CO
(O
o
s-
o
OB
g
8-
3
o
a
CO
1
o
o
s*
<o
fa
D
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It will be seen that the two methods agree for the nine first
places, reckoning from the left ; that beyond this the same written
number, or figure, is read dififerently, or bears different names, in
8 Arithmetic*
in words of a proposed number is formed^ ly reading each division
of figures as if it stood alorie^ and adding^ after its units^ the nanu
belonging to this division.
The above example is read, twenty-four trillions^ eight hundred
and ninety-seven biUions^ thru hundred and twenty-one millions^ five
hundred and eighty thousands^ three hundred and forty-six units.
9. Numbers admit of being considered in two ways ; one is,
when no particular denomination is mentioned, to which their
units belong, and they are then called abstract numbers ; the
other, when the denomination of their units is specified, as when
we say, two men, five years, three hours, &c. these gre called
concrete numbers.
It is evident, that the formation of numbers, by the successive
tinion of units, is independent of the nature of these units, and
that this must also be the case with the properties resulting from
this formation ; by which properties we are enabled to compound
and decompound numbers, which is called calculation. We shall
now explain the principal rules for the calculation of numbers,
without regard to the nature of their units.
Addition.
10, This operation, which has for its object the uniting of
several numbers in one, is only an abbreviation of the formation
of numbers by the successive union of units. If, for instance, it
were required to add five to seven, it would be necessary, in the
the two systems. Seven billiofis, for iDstance, in French is the
same as seven thousands of millions in English. The difference,
however, is odIj in the name ; the written character, and the funda-
mental principles of numeration, are the same in both languages.
The French begin with a new name aflter every three figures, the
English aflter every six. These collections of six, however, are sub-
divided iato portions of three ; so that the Frenfh method is, on the
whole, the more simple, and on this account more easily retained
in the memory, ' while at 4hc same time the names of the higher
orders of numbers are much shorter.
Addiiiotu 9
series of the names of numbers, one^ itooj ihree^ four^ five^ six^
sevm^ &c., to ascend five places above seven, and we should then
come to the word twelve, which is consequently the amount of
seven units added to five. It is upon this process that the addi-
tion of all small numbers depends, the results of which are com-
mitted to memory ; its immediate application to larger numbers
would be impossible, but in this case, we suppose these numbers
divided into the different collections of units contained in them,
and we may add together those of the. same name. For instance,
to add 27 to 32, we add the 7 units of the first number to the 2
of the second, making 9 ; then the 2 tens of the firet with the 3
of the second, making 5 tens. The two results, taken together,
form a total of 5 tens and 9 units, or 59, which is the sum of the
numbers proposed.
What is here said applies to all numbers, however large, that
are to be added together ; but it is necessary to observe, that
the partial]sums, resulting from the addition of two numbers, each
expressed by a single figure, often contain tens, or units of the
next higher collection, and these ought consequently to be joined
to their proper collection.
In the addition of the numbers 49 and 78, the sum of the units
9 and 8 is 17, of which we should reserve 10, or ten, to be added
to the sum of the tens in the given numbers ; next we say that 4
and 7 make 11, and joining to this the ten we reserved, we have
13 for the number of tens contained in the sum of the given
numbers ; which sum, therefore, contains 1 hundred, 2 tens, and
7 units, that is, 1 27.
11. By proceeding on these principles, a method has been de-
vised of placing numbers' that are to be add^, which facilitates
tfie uniting of their collections of units, and a rule has been form-
ed, which the following example will illustrate.
Let the numbers be 527, 2519, 9812, 73, and 8 ; in order to
add them together, we begin by writing them under each other,
placing the units of the same order in the same column ; then
we draw a line to separate them from the result, which is to be
written underneath it.
Arith* 2
10 Arithmtiicm
527
2519
9812
73
8
Sum 12939
We at first find the sum of the numbers contained in the column
of units to be 29, we write down only the nine units, and reserve
the 2 tens, to be joined to those which are contained in the next
column, which, thus increased, contains 13 units of its own order ;
we write down here only the three units, and carry the ten to
the next column. Proceeding with this column as with the
others, we find its sum to be 19 ; we write down the 9 units and
carry the ten to the next column, the sum of which we then find
to be 12; we write down the 2 units under this column and
place the ten on the left of it ; that is, we write down the sum of
this column, as it is found.
By this means we obtain 1 2939 for the sum of the given num-
bers.
12. The rule for performing this operation may be given thus,
Write the numbers to be added under each other ^ so that all the
units of the same kind may stand in the same coltmrn^ and draw a
line under them.
Beginning at the rights add up successively the numbers in each
column ; if the sum does not exceed 9, write it beneath its column^
as it is found ; if it contains one or more tens^ carry them to the
next column ; lastly^ under the last column wriU the vhole of its
t The best method of proving addition is bj means of subtraction.
The learner may, however, in general, satisfy himself of the cor-
rectness of his work by beginning at the top of each column and
adding down, or by separating the upper line of figures and adding
up the rest, and then adding this sum to the upper line.
Subtraction* 11
Examples for practice*
Add together 8635, 2194, 7431, 5063, 2196 and 1225.
Ans. 26734.
Add together 84371, 6250, 10, 3842, and 631 Jlns. 95104.
Add together 3004, 523, 8710, 6345, and 784. Ans. 19366.
Add together 7861, 345, 8023. Ans. 16229.
Add together 66947, 46742, and 132684. Ans. 246373.
Subtraction.
13. After having learned to compose a number by the addi-
tion of several others, the first question that presents itself is,
how to take one number from another that is greater, or, which
amounts to the same thing, to separate this last into two part», one
of which shall be the given number. If, for instance, we have
the number 9, and we wish to lake 4 from il, we should, bj do-
ing this, separate it into two parts, which by addition would be
the same again.
To take one number from another, when they are not large,
it is necessary to pursue a course opposite to that prescribed in
the beginning of article 10, for finding their sum ; that is, in
the series of the names of numbers, we ought to begin from the
greatest of the numbers in question, and descend as many places
as there are units in the smallest, and we shall come to the name
given to the difference required. Ihus, in descending four
places below the number nine we come to Jive^ which expresses
the number that must be added to 4 to make 9, or which shows
how much 9 is greater than 4.
In this last point of view, 5 is the excess of 9 above 4. I( we
ouly wished to show the inequality of the members 9 and 4, with-
out fixing our attention on the order of their values, we should
say that their difference was 5. Lastly, if we were to go through
the operation of taking 4 from 9, we should say that the re-
mainder is 5. Thus we see that, although the words, excess,
remainder, and difference, are synonymous, each answers to r
particular manner of considering the separation of the number 9
into the parts 4 and 5, which operation iis always designated by
the name subtraction.
12 Arithmetic.
14. When the numbers are large, the subtraction is perform-
ed, part at a time, by taking successively from the units of each
order in the greater number, the corresponding units in the
least. That this may be done conveniently, the numbers are
placed as 9587 and 345 in the following example ;
9587
345
Remainder 9242
and under each column is placed the excess of the upper number,
in that column, over the lower, thus ;
5, taken from 7, leaves 2,
4, taken from 8, leaves 4,
3, taken from 5, leaves 2,
and writing afterwards the figure 9, from which there is nothing
to be taken ; the remainder, 9242, shows how much 9587 is
greater than 345.
That the process here pursued gives a trtie result is indisputa-
ble, because in taking from the greater of the two numbers all
the parts of the least, we evidently take from it the whole of the
lieast.
i 5. The application of this process requires particular atten-
tion, when some of the orders of units in the upper number ate
greater than the corresponding orders in the lower.
If, for instance, 397 is to be taken from 534.
524
397
• Remamder 127
In performing this question we cannot at .first take the units
in the lower number from those in the upper; but the number
524, here represented by 4 units, 2 tens, and 5 hundreds, can be
expresssed in a different manner by decomposing some of its col-
lections of units, and uniting a part with the units of a lowef
order. Instead of the 2 tens and 4 units which terminate it, we
can substitute in our minds 1 ten aad 14 units; then taking from
Subtracium* ^13
these units the 7 of the lower numbw , we get the remamder 7*
Bj this decomposition, the upper number now has but one ten,
firom which we cannot take the 9 of the lower number, but from
the 5 hundred of the upper number we can take 1, to join with
the ten that is left, and we shall then have 4 hundreds and 1 1
tens ; taking from these tens the tens t>f the lower number, 2 will
remain. Lastly, taking from the 4 hundreds, that are left in
the upper number, the three hundreds of the lower, we obtain
the remainder 1, and thus get 127 as the result of the operation.
This manner of working consists, as we see, in borrowing,
from the next higher order, an unit, and joining it according to
its value to those of the order, on which we are employed, ob-
serving to count the upper figure of the order from which it was
borrowed one unit less, when we shall have come to it.
16. When any orders of units are wanting in the upper num-
ber, that is, when there are ciphers between its figures, it is
necessary to go to the first figure on the left, to borrow the 10
that is wanted. See an example.
7002
3495
Remainder 3^07 ,
As we cannot take the 5 units of the lower number from the 3
«
of the upper, we borrow 10 units from the 7000, denoted by the
figure 7, which leaves 6990 ; joining ^the 10 we borrowed to the
figure 3, the upper number is now decomposed into 6990 and
13; taking from 13 the 5 units of the lower number, we obtain
7 for the units of the remainder.
This first operation has left in the upper number 6990 units car
699 tens instead of the 700, expressed by the three last figures
on the left ; thus the places of the two ciphers are occupied by
98, and the significant figure on the left is diminished by unity.
Continuing the subtraction in the oth^r colunms in the same
manner, no di£BcuIty occurs, and we find the remainder, as put
down in the example.
17. Recapitulating the remarks made in the two preceding arti«
cles, the rule to be observed in performing subtraction may be
14
Arithmetic*
given thiis. Place the less number under the great^^ so that their
%Lnits of the same order nuiy he in the same column^ and draw a
line under them; heginnit^ at the righty take successioely each
figu,re of the lower number from the, one m the sdme column of the
upper ; if this cannot he done, increase the upper figure by ten itmts^
counting the next significant figure, in the upptr number, less by
unity, and if ciphers come between, regard them as 9s.
18. For greater convenience, when it is nocessary to decrease
the upper figure by unity, we can suSbr it to retain its value, and
add this unit to the corresponding lower figure, which, thus
increased, gives, as is wanted, a result one less than would arise
from the written figures. In the fi^st of the following examples,
after having taken 6 units from 14, we count the next figure of
the lower number 8, as 9, and so in the others.
Examples.
16844
10378
103034
49812002
9786
2437
69845
18924983
7058
33189
173425
8037142
9123724
39742107
67632
5067310
US3467
25378421
Method of proving Addition and Subtractionm
19. In performing an operation, according to a process, the
correctness of which is established upon fixed principles, we may
nevertheless sometimes commit errors in the partial additions
aifd subtractions, the results of which we seek in the memory.
To prevent any mistake of this kind, we have recourse to a
method, the reverse of the first operation, by which we ascertain
whether the results are right; this is called /proving the operation.
The proof of addition consists in subtracting successively from
the sum of the numbers added, all the parts of these numbers,
and if the work has been correctly performed, there will be no
remainder. We will now show by the example given in artkle
11, how to perform all these subtractions at once.
Subtritctum*
627
2519
■ ■ 1
9812
73
•' 1 ■ "i^""
*
8
•
Sum 1 2939
1120
15
We first add the numbers in the left hand column, which here
contains thousands, and subtract the sum 11 from 12, which
begins the preceding result, and write underneath the difference
1, produced by what was reserved from the column of hundreds,
in performing the addition* The sum of the column of hun-
dreds, taken by itself, amounts to but 18 ; if we take this from
the 9 of the first result, increased by borrowing the one thou-
sand, considered as ten hundred, that remains from the column
preceding it on the left, the remainder 1, written beneath, will
show what was reserved from the column of tens. The sum of
the last, 11, taken from 13, leaves for its remainder 2 tens, the
number reserved from the column of units. Joining these 2 tens
with the 9 units of the answer, we form the number 29, which
ought to be exactly the sum of the column of units, as this
column is not affected by any of the others ; adding again the
numbers in this column, we ought to come to the same result,
and consequently to have no remainder. This is actually the
case, as is denoted by the written under the column. The pro*
cess, just explained, may be given thus ; To prove addition^ begin-
mng on the lefi^ add again eadk of ike several columnar subtract the
sums respeciioely from the sums written cAove them and write down
ike remainders which must be joined^ each as so many tens to the
sum of the next column on the right ; if the work he correct there will
be no remainder under the last column^
20. The proof of subtraction is, Ihat the remainder^ added io
tiu less number, exactly gioes the greater* Thus to ascertain the
exactness of the following subtraction, > ^
\
\
^
y.
I& Afiikmtiic. .
%
524.
297
227
524
we add the remainder to the smaller number, and find the smn^
in reality, equal to the greater*
Multiplication.
21 • When the numbers to be added are equal to each other,,
addition takes the name o{ mdlfipUcaiion^ because in this case the
sum is composed of one of the numbers repeated as many times
as there are numbers to be added. Reciprocally, if we wish to
repeat a number several times, we may do it, by adding the
number to itself as many times, wanting one, as it is to be re-^
peated. For instance, by the following addition^
16
16
> 16
16 '-'
64
the numbei' 16 is repeated four times, and added to itself three
times.
To repeat a number twice is to dovhh it ; 3 times, to ttiph it ;
4 times, to qaadrapU it ; and so on.
22. Multiplication implies three numbers, namely, that which
is to be repeated, and which is called the multiplicand ; the num-
ber which shows how many times it is to be repeated, which is
called the mult^lier ; and, lastly, the result of the operation,
which is called the produjf. The multiplicand and mHlt^lier,
considered as concurring io form the product, are called factors
of the product. In the example given above, 16 is the mvit^li-
eand^ 4 the multiplier^ and 64 the product ; and we see that 4 and
16 are the factors of 64.
17
93^ When the multiphcand and multiplier are large numbers,
the formation of the product, by the repeated addition of the
m^iltiplicand, would be very tedious. In consequence of this,
ineans have been sought of abridging it, by separating it into a
certain number of partial operations, easily perfoi^med by mem*
ory. For instance, the number 16 would be repeated 4 times,
by taking separately, the' same number of times, the six units and
the ten, that compose it. It is sufficient, then, to know the pro-
ducts arising from the multiplication of the units of each order
in the multiplicand by the multiplier, when the multiplier consists
of a single figure, and this amounts, for all cases that can occur,
to finding the products of each one of the 9 first numbers by every
other of these numbers.
34. These products are contained in the following table, attri-
buted to Pythagoras.
*BJHMLB CM* CSMUUBCBbMa
1
3
3
4
5
6
7
8
9
4
6
8
12
10
15
99
12
14
16
18
s
6
a
9
18
21
24
27
4
5
6
7
«
18
20
24
28
32
36
10
12
14
16
18
15
18
25
30
35
40
45
24
28
30
36
42
48
54
21
24
27
35
42
49
63
72
8
32
36
40
48
56
64
9
45
54
63
72
81
35. To form this table, the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9,
are written first on the same line. Each one of these numbers
is then added to itself and the sum written in the ^second line,
which thus contains each number of the first doubled, or the
Arith. 3
nr ArUkoxtik*
product of each number by 2. Each number of the secbnd Ime'
is then added to the number over it in the first, and their sum?
are written in the third line, which thus contains the triple of
each number in the first, or their products by 3. By adding the
numbers of the third line to those of the first, a fourth is formed,
containing the quadruple of each number of the first, or their
products by 4 ; and so on to the ninth line, which contains the
products of each number of the first liAe by 9*
It may not be amiss to remark, that the difierent products of
any number whatever by the numbers 2, 3, 4, 5, &c. are called
multiples of that number ; thus, 6, 9, 13, 15, &c. are multiples of 3.
36. When the formation of this table is well understood, the
mode of using it may be easily conceived. If, for instance, the
product of 7 by 5 were required ; looking to the fifth line, which
contains the difierent products of the 9 first numbers by 5, we
should take the one directly under the 7, which is 35 ; thesame
method should be pursued in every other instance, and iht pro^
duct will always be found in the line of the multiplier and under
the multiplicand*
27, If we seek in the table of Pydiafgoras' the product of 5 by
7, we shall find, as before, 35, although in this case 5 is the mul-^
tiplicand, and 7 the multiplier. This remark is applicable to each
product in the table, and it is possiblej in any multiplication, to
reverse the order of the factors ; that is, to make the multiplicand
the mtUtiplier, and the mult^lier the multiplicand.
As the table of Pythagoras contains but a limited number of
products, it would not be sufficient to verify the above conclu-^
sion by this table; for a doubt might arise respecting it in the
case of greater products, the number of which is unlimited ;
there is but one method independent of the particular value of *
the multiplicand and multiplier of showing that there is no ex-
ception to this remark. This is one well calculated for the pur-
pose, as it gives a good illustration of the manner, in which the
product of two numbers is formed. To make it more easily un-
derstood, we will apply it first to the factors 5 and 3.
If we write the figure 1 five times on one line, and place tw»
similar lines underneath the first, in this manner.
MtUt^lkatimt. 49
1, 1, 1, 1, 1,
1, 1, 1, 1, 1,
1, 1, 1, 1, 1,
lihe whole number of Is will consist of as many times 5 ds there
lines, that is, 3 times 5 ; but, by the disposition of these lines, the
.figures are ranged in columns, containing 3 each. Counting them
in this manner, we jSnd as many times 3 units as there are col-
jumns, or 5 times 3 units, and as the product does not depend on
the manner of counting, it follows that 3 times 5 and 5 times 3
give the same product. It is easy to extend this reasoning to
any numbers, if we conceive each line to contain as many units
as there are in the multiplicand, and the number nf Iine;s, placed
one under the other, to be equal to the multiplier. In counting
the product by lines, it arises from the ipultiplicandirepe^ted as
many times as there are units in the multiplier; but the asscm-
blag^of figures written presents as many columns as there are
units in e line, and each column contains as many units as
as there are lines ; if, then, we choose to count by columns, the
number of lines, or the multiplier, will be repeated as many
times as there are units^ in a line, that is, in the multiplicand. We
may therefore, in finding the product of any two frambers, take
-either of them at pleasure, for the multiplier.
28. The reasoning, just given to prove the truth of the pre-
ceding proposition, is the demonstration of it^ and it may be
remarked, that the essential distinction of pure mathematics is,
that no proposition, or process, is admitted, which is not the
necessary consequence of the -primary notions on which it is
founded, or the truth Qf wbiQh is not generally established by
reasoning independent of particular examples, which can neVer
constitute a proof, but serve only to facilitate the reader's under^
standing the reasoning,, or the practice of the rules.
29; Knowing all the products given by the nine first number^
combined with each other, we can, according to the remark in
article 23, multiply any number by a number consisting of a
single figure, by forming successively the product of each order
of units in the multiplicand, by the multiplier; the work is aB
.firflDws ;
M Jlrifhmlit*
536
7
3682
The product of the units of the multiplicand, 6, by the multi-
plier, 7, being 42, we write down only the 2 units, reserving the
4 tens to be joined with those that will be found in the next
higher place.
The product of the tens of the multiplicand, 2, by the multi-
plier, 7, is 1 4, and adding the 4 tens we reserved, we make them
] 8, of which number we write only the units, and reserve the ten
for the next operation.
The product of the hundreds of the multiplicand, 5, by the
multiplier, 7, is 35 ; when increased by the 1 we reserved, it
becomes 36, the whole of which is written, because there are
no more figures in the multiplicand. |^
80. This process may be given thus ; To multiply a ntctnber
of several figures by a single figure^ place the multiplier under the
units of the multiplicand^ and draw a line beneath^ to separate them
from the product* Beginning at the figkt^ multiply successvoely^ iy
the multiplier J the units of each order in the multiplicand^ and write
the whole product ofeach^ when it does not exceed 9 ; but, if it covif'
tains tens^ reserve them to be added to the next pnJduct. Continue
thus to the last figure of the multiplicand, on the left, the whole result
of which must be written down*
Examples. 243 by 6. Ans. 1458. 8943 by 9. Ans. 80487.
It is evident that, when the multiplicand is terminated by Os,
the operation can commence only with its first significant figure;
but to give the product its proper value, it is necessary to put,
on the right of it, as many Os as there are in the multiplicand.
As for the Os which may occur between the figures of the mul«
tiplicand, they give no product, and a must be written down
when no number has been reserved from the preceding product,
as is shown by the following examples :
956 8200 7012 80970
6 9 5 4
5736 73800 35060 323880
• MuUiplicaHm. 21
Mulliplj
730 by 3. Ans. SI 90. 8104 by 4. Am. 39416.
20508 by 5. Ans. 102540. 360500 byC j$n«. 2163000.
297000 by 7. Ans. 2079000. 9097030 by 9. Ans. 81873270.
31. The most simple number, expressed by several figureft|
being 10, 100, 1000, &c., it seems necessary to inquire how we
can multiply any number by one of these. Now if we recollect
the principle mentioned in article 6, by which the same figure is
increased in value 10 times, by every remove towards the left,
we shall soon perceive, that to multiply any number by 10, we
must make each of its 6rders of units ten times greater ; that is,
we must change its units into tens, its tens into hundreds, and
so on, and that this is effected by placing a on the right of the
number proposed, because then all its significant figures will be
advanced one place towards the. left.
For the same reason, to multiply any number by 100, we
should place two ciphers on the right ; for, since it becomes ten
times greater by the first cipher, the second will make it ten
times greater still, and consequently it will be 10 times 10, or
100 times, greater than it was at first.
Continuing this reasoning, it will be perceived that, according
to our system of numeration, a number fs nmltiplied by 10, 100,
1000, &c. by writing on the right of the multiplicand as many
ciphers as there are on the right of the unit in the multiplier.
32. When the significant figure of the multiplier differs from
unity, as, for instance, when it is required to multiply by 30, or
300, or 3000, which are only 10 times 3, or 100 times 3, or 1000
times 3, 8tc. the operation is made to consist of two parts ; we at
first multiply by the significant figure, 3, according to the rule
in article 30, and then multiply the product by 10, 100, or 1000,
^cv (as was stated in the preceding article) by writing one, two,
Abree, &c., ciphers on the right of this product.
.: Let it be required, for instance, to multiply 764 by 300.
764
300
229200
22 Arithmeilc.
The four significant figures of this product result from the
multiplication of 764 by 3, and are placed two places towards
the left ^o admit thdfttwo ciphers, which terminate the multiplier.
In general, zohen ihe multiplier is terminated by a number of
ciphers^ first multiply the multiplicand btf the significant figure of
the multiplier J and place^ after the product, as many ciphers as thtrp
are in the multiplier*
Examples*
Multiply
35012 by 100. .4m. 3501200. 638427 by 500. Ans. 319213500;
2107900 by 70. Ans. 147553000. 9 1 20400 by 90.. ^iw. 82083600a.
33. The preceding rules apply to the case, in which the mul-
tiplier is any number whatever, by considering separately each
of the collections of units of which it is composed. To multiply,
for instance, 793 by 345, or, which is the same thing, to repeat
703, 345 times, is to take 793, 5 times, added to 40 times, added
to 300 times, and the operation to be performed is resolved into
3 others, in each of which the multipliers, 5, 40, and 300, have
but one significant figure.
To add the result of these three operations easily, the calculav
tion is di^ioscd thus ^
793
345
3965
31720
237900 •
2735351
The multiplicand is multiplied successivety by the units, tenss
hundreds, &c. of the multiplicrj observing to place a cipher ok
the right of the partial product, given by the tens in the multi-
plier, and two on the right of the product givea by hundreds,
which advances the first of these products one place towards the
left, and the second, two. The three partial products are thea
added together, to obtain the total product of the given numbers*
Afuliiplkatuni^ 2$
As the ciphers placed at the end of these partial products, are
©f no value in the addition, we may dispense with writing them,
provided we take care to put in its proper place the first figure
of the product given by each significant figure of the multiplier ;
that is, to put in the place of tens the first figure of the product
given by the tens in the multiplier ; in the place of hundreds the
first figure of the product given by the hundreds in the multiplier,
and so on.
34. According to what has been said, the rule is as follows.
To multiply any two numbers^ one by the other^ form successively
{according to the rule in article 30,) the products of the multipli-
cand^ by the different orders of units in the multiplier ; observing to
place the first figure of each partial product under the units of the
same order with the figure of the multiplier j by which the product is
given; and then add together all the partial products*
35. When the multiplicand is terminated by ciphers, they may
at first be neglected, and all the partial multiplications begin
with the first significant figure of the multiplicand ; but after-
wards, to put in their proper rank the figures of the total product,
as many ciphers, as there are in the multiplicand, must be writ-
ten on the right of this product.
If the multiplier is terminated by ciphers, we may, according
to the remark in article 31, neglect these also, provided we write
an equal number on the right of the product*
Hence it results that, when both multiplicand and multiplier are
ierfninated by ciphers^ these ciphers may at first be neglected, and
afUr the other figures of the product are obtained, the same number
may be written on the right of the product.
When there are ciphers between the significant figures of the
multiplier, as they give no product, they may be passed over,
observing to put in its proper place the unit of the product, given
by the figure on the left of these ciphers.
24 ArMmetic,
SOO 536 Multiply 9648 by 5137. Ans'. 49561776,
40 307 7854 by 350. Ans. 2748900$.
17204774 by 125. Ans. 2150596750.
12000 3682 62500 by 520. Am. 32500000.
157800 25980762 by 40. Ans. 1039,230480.
161482.
Division.
36. The product of two numbers being formed by repeating
one of these numbers as many times as there are units in the
other, we can, from the product, find one of the factors, by ascer^
taining how many times it contains the other ; subtraction alone
is necessary for this. Thus, if it be required to ascertain the
number of times 64 contains 16, we need only subtract 16 frbm
64 as many times as it can be done ; and since, after 4 subtrac-
tions, nothing is left, we conclude, that 1 6 is contained 4 times in
64. This manner of decomposing one number by another, in
order to know how many times the last is contained in the first,
is called dtpmon, because it serves to divide, or portion out, a
given number into equal parts, of which the number or value is
given.
If, for instance, it were required to divide 64 into 4 equal
parts ; to find the value of these parts, it would be necessary to
ascertain the number, that is contained 4 times in 64, and conse-
quently to regard 64 as a product, having for its factors 4 and
one of the required parts, which is here 16.
If it were asked how many parts, of 16 each, 64 is composed
of, it would be necessary, in order to ascertain the number of
these parts, to find how many times 64 contains 16, and conse**
quently, 64 must be regarded as a product, of which one of the
factors is 16, and the other the number sought, which is 4.
Whatever then may be the object in view, division consists in
finding one of the factors of a given product^ when the other is knovm.
Dwision* 25
37. The number to be divided is called the dividend ; the fac-
tor, that is known, and by which we must divide, is called the
divisor ; the factor found by the division is called the quolitnt^
and always shows how many times the divisor is contained in
the dividend.
It follows then, from what has been said^ that the divisor mul-
tiplied by the quotient must reproduce the dividend.
38. When the dividend can contain the divisor a great many
times, it would be inconvenient in practice to make use of repeated
subtraction for finding the quotient ; it then becomes necessary
to have recourse to an abbreviation analogous to that which is
given for multiplication. If the dividend is not ten times larger
than the divisor, which may be easily perceived by the inspec-
tion of the numbers, and if the divisor consists of only one figure,
the quotient may be found by the table of Pythagoras, since that
contains all the products of factors* that consist of only one
figure each. If it were asked, for instance, how many times 8 is
contained in 56, it would be necessary to go down the 8th column,
to the li^e in which 56 is found ; the figure 7, at the beginning
of this line, shows the second factor of the number 5S^ or how
many times 8 is contained in this number.
We see by the same table, that there are numbers, which can*
not be exactly divided by others. For instance, as the seventh
line, which contains all the multiples of 7, has not 40 in it, it
follows that 40 is not divisible by 7 ; but as it comes between 35
and 42, we see that the greatest multiple of 7, it can contain, is
35, the factors of which are 5 and 7. By means of this ele-
mentary information, and the considerations which will now be
offered, any division whatever may be performed,
39. Let jt be required, for example, to divide 1656 by 3 ; this
question may be changed into another form, namely ; To find
such a number^ that multiplying its units^ tens^ hundreds^ ^c. by 3,
the product of these tint/;, tens^ hundreds^ ^c. may be the dividetidj
1656. jpr
It is plain, th^ this number will not have units of a higher
order than thousands, for, if it had tens of thousands, there
would be tens of thousands in the product, which is not the case.
Arith. 4
26 Arithmetic.
Neither can it have units of as high an order as thousands, for if
it had but one of this order, the product would contain at least 3,
which is not the case. It appears then, that the thousand in the
dividend is a number reserved, when the hundreds of the quo-
tient were multiplied by 3, the divisor.
This premised, the figure occupying the place of hundreds, in
the required quotient, ought to be such, that, when multiplied by
3, its product may be 16, or the greatest multiple of 3 less than
16. This restriction is necessary, on account of the reserved
numbers, which the other figures of the quotient may furnish,
when multiplied by the divisor, and which should be united to
the product of the hundreds.
The number, which fulfils this condition, is 5 ; but 5 hundreds,
multiplied by 3, gives 15 hundreds, and the dividend, 1656, con-
tains 16 hundreds; the difference, 1 hundred, must ^ave come
then from the reserved number, arising from the multiplication of
the other figures of the quotient by the divisor. If we now sub-
tract the partial product, 15 hundreds, or 1500, from the total
product, 1656, the remainder, 156, will contain the product of
the units and tens of the quotient by the divisor, and the question
will be reduced to finding a number, which, multiplied by 3, gives
156, a question similar to that which presented itself above.
Thus when the first figure of the quotient shall have been found
in this last question, as it was in the first, let it be multiplied by
the divisor ; then subtracting this partial product from the whole
product, the result will be a new dividend, which may be treated
in the same manner as the preceding, and so on, until the origi*
nal dividend is exhausted.
40. The operation just described is disposed of thus ;
dividend 1656
3
divisor
15
652
quotient
15
15
06
6
Dhisum, 37
The dividend and divisor are separated by a line, and another
line is drawn under the divisor, to mark the place of the quotient.
This being done, we take on the left of the dividend the part 16,
capable of containing the divisor, 3,* and dividing it by this num-
ber, we get 5 for the first figure of the quotient on the left ; then
taking the product of the divisor by the number just found, and
subtracting it from 16, the partial dividend, we write, under-
neath, the remainder, 1, by the side of which we bring down the
5 tens of the dividend. Considering the number, as it now
stands, a second partial dividend, we divide it also by the divi-
sor, 3, and obtain 5 for the second . figure of the quotient ; we
then take the product of this number by the divisor, and subtract-
ing it from the partial dividend, get for the remainder. We
then bring down the last figure of the dividend, '6, and divide
this third partial dividend by the divisor, 3, and get 2 for the .
last figure of the quotient.
41. It is manifest that, if we find a partial dividend which can-
not contain the divisor, it must be because the quotient has no
units of the order of that dividend, and that those which it con-
tains arise from the products of the divisor by the units of the
lower orders in the quotient ; it is necessary, therefore, when-
ever this is the case, to put a in the quotient^ to occupy the
place of the order of units that is wanting.
For instance, let 1535 be divided by 5.
1535
5
15
307
035
SB
00
The division of the 15 hundreds of the dividend, by the divi«
sor, leaving no remainder, the 3 tens, which form the second par-
tial dividend, do not contain the divisor. Hence it appears, that
the quotient ought to have no tens ; consequently this place must
be filled with a cipher, in order to give to the first figure of the
quotient the value it ought to have, compared with the others .;
38 Ariihmtiic.
then bringing down the last figure of the dividend, we form a
third partial dividend, which, divided by 5, gives 7 for the units
of the quotient, the whole of which is now 307.
42. The considerations, presented in article 40, apply equally
to the case, in which the divisor consists of any number of
figures.
If, for instance, it were required to divide 57981 by 251, it
would easily be seen, that the quotient can have no figures of a
higher order than hundreds, because, if it had thousands, the divi-
dend would contain hundreds of thousands, which is pot the case ;
further, the number of hundreds should be such, that, multiplied
by 251, the product would be 579, or the multiple of 251 next
less than 579 ; this restriction is necessary on account of the
reserved numbers which may have been furnished by the multi-
plication of the other figures of the quotient by the divisor. The
number, which answers to this condition, is 2; but 2 hundreds,
multiplied by 251, give 502 hundreds, and the divisor contains
579 ; the difference, 77 hundreds, arises then from the reserved
numbers resulting from the multiplication of the units and tens of
the quotient, by the divisor.
If we now subtract the partial product, 502 hundreds, or
50200, from the total product, 57981, the remainder, 7781, will
contain the products of the units and tens of the quotient by the
divisor, and the operation will be reduced to finding a number,
which, multiplied by 251, will give for a product 7781.
Thus, when the first figure of the quotient shall have been de-
termined, it must be multiplied by the divisor ; the product being
subtracted from the whole dividend, a new dividend will be the
result, which must be operated upon like the preceding ; and so
on, till the whole dividend b exhausted.
It is always necessary, for obtaining the first figure of the
quotient, to separate, on* the left of the dividend, so many figures,
as, considered as simple units, will contain the divisor, and admit
of this partial division.;
43. Disposing of the operation as before, the calculation, just
explained, is performed in the following order ;
JDhuion*
S9
57981
251
602
231
778
753
251
251
000
The 3 first figures, on the left of the dividend, are taken to
form the partial dividend ; they are divided bj the divisor, and
the number 2, thence resulting, is written in the quotient ; the
divisor is then multiplied by this number, and the product, 502,
is written under the partial dividend, 579. Subtraction being
perfcHrmed, the 8 tens of the dividend are brought down to the
side of tbe remainder, 77 ; this new partial dividend is then
divided by the divisor, and 3 is oUained for the second figure of
the quotient ; the divisor is multiplied by this, the product sub-
tracted from the corresponding partial dividend, and to the re-
mainder, 25, is brought down the last figure of the dividend, 1 ;
this last partial dividend, 251, being equal to the divisor, gives 1
for the units of the quotient.
44. When the divisor contains many figures, some difficulty
may be found in ascertaining how many times it is contained in
the partial dividends. The following example is designed to
show how it may be known.
423405
3880
485
873
3540
3395
1455
1455
0000
30 Arithmetic.
It is necessary at first to take four figures on the left of the
dividend, to form a number which will contain the divisor; and
then it cannot be immediately perceived how many times 485 is
contained in 4234. To aid us in this inquiry, we shall observe,
that this divisor is between 400 and 500 ; and if it were exactly
one or the other of these numbers, the question would be reduced
to finding how many times 4 hundred or 5 hundred is contained
in the 42 hundreds of the number 4234, or, which amounts to the
same thing, how many times 4 or 5 is contained in 42. For the
first of these numbers we get 10, and for the second 8 ; the quo-
tient must now be sought between these two. We see at first
that we cannot eofploy 10, because this would imply, that the
order of units in the dividend above hundreds contained the
divisor, which is not the case. It only remains, then, to try
which of the two numbers 9 or 8, used as the multiplier of 485,
gives a product that can be subtracted from 4234, and 8 is found
to be the one. Subtracting from the partial dividend the pro-
duct of the divisor multiplied by 8, we get, for the remainder,
354 ; bringing down then the tens in, the dividend, we form a
gecond partial dividend, on which we operate as on the preced-
ing ; and so with the others.
45. The recapitulation of the preceding article gives us this
rule. To divide one number by another^ place the divisor on the
right of the dividend^ separate them by a /tne, and draw another
line under the divisor, to make the place for the quotient. Take, on
the left of the dividend, as many figures as are necessary to contain
the divisor ; find ham many times the number expressed by the first
figure of the divisor, is contained in that, rq^resented by the first figure
or two first figures of the partial dioidend; multiply this quotient,
iohich is only an approximation, by the divisor, and, if the product
is greater than the partial dividend, take units from the quotient
contimially, till it will give a product that can be subtracted from
the partial dividend ; subtract this product, and if the remainder
be greater than the divisor, it will be a proof that the quotient has
been too much diminished ; and, consequently, it must be increased.
By the side of the remainder bring down the next figure of the
dividend, and find, as before, how many time^ this partial dividend
ctmtains the divisor ; continue thus, until all the figures of the given
Division* . 31
dividend are brought down* When a partial dividend occurs^ which
does not contain the divisor^ it is necessary, before bringing dozen
another figure of the dividend, to put a cipher in the quotient*
46. The operations required in division may be made to oc-
cupy a less space, by performing mentally the subtraction of the
products given by the divisor and each figure of the quotient, as
is exhibited b the following example ;
39
1755
195
000
45
After having found that the first partial dividend contains 4
times the divisor, 39, we multiply at first the 9 units by 4, which
gives 36 ; and, in order to subtract this product from the partial
dividend, we add to the 5 units in the dividend 4 tens, making
their sum 45, from which taking 36, 9 remains. We then re-
serve 4 tens to join them, in the mind, to 12, the product of the
quotient by the tens in the divisor, making the sum 16 ; in taking
this sum from 1 7, we take away the 4 tens, with which we had
augmented the units of the dividend, in order to perform the
preceding subtraction. We then operate jn the same manner on
the second partial dividend, 195, saying; 9 times 5 make 45,
taken from 45, nought remains ; then 5 times 3 make 15, and 4
tens, reserved, make 19, taken fron^ 19, nought remains.
We see sufficiently by this in what manner we are to perform
any other example, however complicated.
47. Division is also abbreviated when the dividend and divisor
are terminated -by ciphers, because we can strike out, from the
end of each, as many ciphers as are contained in the one that
has the least number.
If, for instance, 84000 were to be divided by 400, these num-
bers may be reduced to 840 and 4, and the quotient would not
be altered ; for we should only have to change the name of the
units, since, instead of 84000, or 840 hundreds, and 400, or 4
hundreds, we should have 840 units and 4 units, and the quotient
of the numbers 840 and 4 is always the same, whatever may be
the denomination of their units.
32
Arithmetic.
It may also be remarked that, in striking out two ciphers at
the end of the given numbers, they have been, at the same time,
both of them divided by 100; for it follows from article 31, that
in striking out 1 , 2, or 3 ciphers on the right of any number, the
number is divided by 10, or 100, or 1000, &c.
Examples in Division.
144
24
00
48
16512
2752
0000
344
48
3049164
53956
37644
00000
6274
486
Divide 49561776 by 6137.
27489000 by 350.;
2160596750 by 125.
32500000 by 6i0.
10392304800 by 9^'
Ans. 9648.
Ans. 7854.
Ans. 17204774.
Ans. 62500.
Ans. 25980762.
48. Division and multiplication mutually prove each other,
like subtraction and addition : for, according to the definition of
division (36), we ought, by dividing the product by one of the
factors, to find the other, and multiplying the divisor by the
quotient, we ought to reproduce the dividend (37).
Fractions.
49. DiYisioN cannot always be performed, so as not to leave
a remainder, because every number of units is not exactly com**
posed of any other number whatever of units, taken a certain
number of times. Examples of this have already been seen in
the table of Pythagoras, which contains only the product of the
9 first numbers multiplied two and two, but does not contain all
the numbers between 1 and 81, the first and last numbers in it.
The method hitherto given shows, then, only how to find the
greatest multiple of the divisor^ that can be contained in the
dividend.
If we divide 239 by 8, according to the rule in article 46,
a-
239
79
7
8
29
we have for the last partial dividend, the number 79, which does
not contain 8 exactly, but which, falling between the two numbers,
72 and 80, one of which contains the divisor, 8, nine times, and
the other ten, shows us that the last part of the quotient is greater
than 9, and less than 10, and consequently, that the whole quo-
tient is between 29 and 30. If we multiply the unit figure of
the quotient, 9, by the divisor, 8, and subtract the product from
the last partial dividend, 79, the remainder, 7, will evidently be
the excess of the dividend, 239, above the product of the factors,
29 and 8. Indeed, having, by the different parts of the operation,
subtracted successively from the dividend, 239, the product of
each figure of the quotient by the divisor, we have evidently sub-
tracted the product of the whole quotient by the divisor, or 232 ;
and the remainder, 7, less than the divisor, proves, that 232 is
the greatest multiple of 8, that can be contain^ in 239.
50. It must be perceived, after what has been said, that to
reproduce any dividend, we must add to the product of the divi-
sor bjr the quotient, the. sum which remains when the division
cannot be performed exactly.
«
51. If we wished to divide into eight equal parts a sum of
whatever nature, consisting of 239 units, we could not do it with-
out using parts of units or fractions. Thus, when we have taken
from the number 239 the 8 times 29 units contained in it, there
will remain 7 units, to be divided into 8 parts ; to do this, we
may divide each of these units, one after the other, into 8 parts,
and then take one part out of each unit, which will give 7 parts
to be joined to the 29 whole units, to form the eighth part of 239,
or the exact quotient of this number, by 8.
The same reasoning may be applied to every other example
of division- in which there is a remainder, and in this case the
quotient is composed of two parts ; one, consisting of whole units,
while the other cannot be obtained until the concrete or material
units of the remainder have been actually divided into the num*
Jlriih. 5
84 Ariihmetic.
ber of parts denoted by the divisor ; without this it can only
be indicated by supposing, a unit of the dividend to be divided
into as many parts as there are units in tlu divisor^ and so many of
these partSj as there are units in the remainder^ taken to complete the
quotient required*
52. In general, when we have occasion to consider quantities
less than unity, we suppose unity divided into a certain number
of parts sufficiently small to be contained a certain number of
times in these quantities, or to measure them. In the idea thus
formed of their magnitude there are two elements, namely, the
number of times the measuring part is contained in unity, and
the number of these parts found in the quantities.
A nomenclature has been made for fractions, which answers
to this mapner of conceiving and representing them.
That which results from the division of unity
into 2 parts is called a moiety or half
into 3 parts a thirds
into 4 parts a quarter or fourth^
into 5 parts ^f\f^
into 6 parts a sixth,
and so on, adding after the two first, the termination th to the
number, which denotes how many parts are supposed to be m
unity.
Every fraction then is expressed by two numbers ; the first,
which shows how many parts it is composed of, is called the
numeratorj and the other, which shows how many of these parts
are necessary to form an unit, is called the denominatory because
the denomination of the fraction b deduced from it. Five sixths
of an unit is a fraction, the numerator of which is fve, and the
denominator six*
The numerator and the denominator together are called the two
terms of the fraction.
Figures are used to shorten the expression of fractions, the
denominator being written under the numerator, and separated
from it by a line,
one third is written <|,
Jhe sixths !*•
Fractiom* $5
53. According to the meaning attached to the words, numeror
tor and denominator^ it is plain, that a fraction is increased^ by
increasn^ its numerator without changing its denominator ; for
this last, as it shows bto how many parts unity is divided, deter-
mines the magnitude of these parts, which continues the same,
while the denominator remains unchanged ; and by augmenting
the numerator, the number of these parts is augmented, and con-
sequently the fraction increased. It is thus, for instance, that ^
exceeds ^, and that ^^ exceeds |^.
It follows evidently from this, that hy repeating the numerator
% 3, or any number of titius^ without alterir^ the denominator, we
repeat, a like nujnher of times, the quantity expressed by the fraction,
or in other words multiply it by this number ; for we make 2, 3, or
any number of times, as many parts, as it had before, and these
parts have remained each of the same value.
The fraction |, then, is the triple of \ and W the double o{{^»
A fraction is diminished by diminishing its numerator, without
that^ging its denominator, since it is made to consist of a less
number of parts than it contained before, and these parts retain
the same value. Whence, if the numerator be divided by 2, 3, or
any number, without the denominator being altered, the fraction is
made a like number of times smaller, or is divided by that number,
for it is made to contain % 3, or any number of times less parts
than it contained before, and these parts remain of the same
yalue. Thus | is a third of | and ^'j is half of ^f.
54. On the contrary, a fraction is diminished, when its de-
nominator is increased without changing its numerator ; for then
more parts are supposed in an unit, and consequently they must
be smaller, but, as only the same number of them are taken to
form the fraction, the amount in this case must be a less quan-
tity than in the first. Thus | is less than f , and -^ than f •
Hence it follows. Chat if the denominator of a fraction be multi'
plied by % 3, or any number, without the numerator 6etng changed,
Refraction becomes a like number of times smaller, or is divided by
that number, for it is composed of the same number of parts as
|)efore, but each of them has becomes 2, 3, or a certain number
pf tiiaes less» The fraction ^ is half of |, and j\ the third of f.
SS ' Arithmetic.
A fraction is increased when its denominator is diminished with-
ihit the numerator being changed ; because, as unity is supposed to
be divided into fewer parts, each one becomes greater, and their
amount is therefore greater.
Whence, if the denominator qfa fraction be divided 6y 2, 3, or
any other number^ the fraction will be made a like number of times
greater, or will be multiplied by that number ; for the number of
parts remains the same, and each one becomes 2, 3, or a certain
number of times greater than it was before. According to this,
I is triple of -^ and j- the quadruple of /^.
It may be remarked, that to suppress the denominator of a
fraction is the same as to multiply the fraction by that number.
For instance, to suppress the denominator 3 in the fraction y is to
change it into 2 whole ones, or to multiply it by 3.
55. The preceding propositions may be recapitulated as fol-
lows;
B^SSrl .he „„„«,»», A. &,«»„ » \^^-^
56. The first consequence to be drawn from this table is, that
the operations performed on the denominator produce effects of
an inverse or contrary nature with respect to the value of the
fraction. Hence it results, that^ if both the numerator and denomr
inaior of a fraction be multiplied at the same time, by the same
number, the value of tfu fraction will not be altered ; for if, on the
one hand, multiplying the numerator makes the fraction 2, 3, &c.
times greater, so on the other, by the second operation, the half
or third part, &c. of it is taken ; in other words, it is divided by
the same number, by which it had at first been multiplied.
Thus I is equal to ^^y, and -^-^ is equal to i|.
57. It is also manifest that, if both the numerator and denomi-
nator of a fraction be divided, at the same time, by the same «tim-
ber, the value of the fraction will not be altered ; for if, on the one
hand, by dividing the numerator the fraction is made 2, 3, &c.
times smaller ; on the other, by the second operation, the doubloi
FractiofiUn $i
triple, &c. is taken ; in short it is multiplied by the same num-
ber, hj which it was at first divided. Thus the fraction | i»
equal to |, and | is equal to |.
58. It is not with fractions as with whole numbers, in which a
magnitude, so long as it is considered with relation to the same
unit, is susceptible of but one expression. In fractions on the
contrary, the same magnitude can be expressed in an infinite
number of ways. For instance, the fractions
it a 4 f « 7 ^r
» f ? ri T? IZt TJJ TT' ^^*
in each of which the denominator is twice as great as the nume-
rator, express,. under difierent forms the half of an unit. I'he
fractions
11*4 8 9 f ^r
Ti T9 T? T¥? TT9 TT? TU ^^*
of which the denominator is three times as great as the numera-
tor, represent each the third part of an unit. Among all the
forms, which the given fraction assumes, in each instance, the
first is the roost remarkable, as being the most simple ; and, con-
sequently, it is well to know how to find it from any of the others.
It is obtained by dividing the two terms of the others by the
same number, which, as has ah'eady been shown, does not alter
their value. Thus if we divide by 7 the two terms of the frac-
tion 1-^:^, we come back to ^ *, and, performing the same opera-
tion on f\, we get •}•
59. It is by following this process, that a fraction is reduced
most simple terms ; it cannot, however, be applied, except to
fraction^V^pf which the numerator and denominator are divisible
by the sam^^umber ; in all other cases the given fraction is the
most simple of all those, that can represent the quantity it ex-
presses. Thus the fractions 4? tt? if? ^^ terms of which can-
not be divided by the same number, or home no common divisor^
are irreducible^ and, consequently, cannot express, in a more sim-
ple manner, the magnitudes which they represent.
60. Hence it follows, that to simplify a fraction, we must
endeavour to divide its two terms by some one of the numbers,
3, 3, &c. ; but by this uncertain mode of proceeding it will not
38 Arithmetic.
be always possible to come at the most sitnpU terms of the given
fraction, or at least, it will often be necessary to perform a great
Dumber of operations*
If, for instance, the fraction }$ were given, it may be seen at
^ once, that each of its terms is a multiple of 2, and dividing them
by this number, we obtain ^| ; dividing these last also by 2, we
obtain /j. Although much more simple now than at first, this
fraction is still susceptible of reduction, for its two terms can be
divided by 3, and it then becomes ^.
If we observe, that to divide a number by 2, then the quotient
by 2, and then the second quotient by 3, is the same thing as to
divide the original number by the product of the numbers, 2, 2,
and 3, which amounts to 12, we shpU see that the three above
operations can be performed at once by dividing the two terms
of the given fraction by 12, and we shall again have ^.
The numbers 2, 3, 4, and 12, each dividing the two numbers
S4 and 84 at the same time, are the common divisors of these
numbers; but 12 is the most worthy of attention, because it is
the greatest, and it is by employing the greatest common divisor
4>f the two terms of the given fraction, that it is reduced at once
to its most simple terms. We have then this important problem
to solve, tpo numbers being given^ to find their greatest common
divisott*
•^
61. We arrive at the knowledge of the common divisor of two
numbers by a sort of trial easily made, and which has this re-
commendation, that each step brings us nearer and nearer to
the number sought* To explain it clearly, I will take an ex-
ample.
Let the two numbers be 637 and 143. It is plain, that the
greatest common divisor of these two numbers cannot exceed the
smallest of them ; it is proper then to try if the number 143)
which divides itself and gives 1 for the quotient, will also divide
the number 637, in which case it will be the greatest common
divisor sought. In the given example this is not the case ; we
obtain a quotient 4, and a remainder 65.
■ ' ^ l» I II «l—l ■ II ■ I » I ■■ I ll.lli.. ■ I ■ I n il — P^-l III ■ ■
t What is here called the greatest common divisor^ is sometimes
called the greaiut common measure.
Praciicns. 39
Now it is plain, that every common divisor of the two num-
bers, 143 and 637, ought also to divide 65, the remainder result-
ing from their division ; for the greater, 637, is equal to the less,
143, multiplied by 4, plus the remainder, 65, (50); now in
dividing 637 by the common divisor sought, we shall have an
exact quotient ; it follows then, that we must obtain a like quo-
tient, by dividing the assemblage of parts, of which 637 is com-
posed, by the same divisor ; but the product of 1 43 by 4 must
necessarily be divisible by the common divisor, which is a factor
of 143, and consequently the other part, 65, must also be divisi-
ble by the same divisor ; otherwise the quotient would be a whole
number accompanied by a fraction, and consequently could not
be equal to the whole number, resulting from the division of 637
by the common divisor. By the same reasoning, it may be
proved in general, that every common divisor of two numbers must
also divide the remainder resultit^ from the division of the greater
of the two by the less.
According to this principle, we see, that the common divisor
of the numbers 637 and 143, must also be the common divisor
of the numbers 143 and 65 ; but as the last cannot be divided by
a number greater than itself, it is necessary to try 65 first.
Dividing 143 by 65, we find sJ quotient 2, and a remainder 13;
65 then is not the divisor sonrght. By a course of reasoning,
similar to that pursued wilh^,rc^rd to the numbers, 637, 143,
and the remainder, resulting jfom t|ieir division, 65, it will be seeii
that every common divisor of /1 43 and 65 must also divide the
numbers 65 and 13; now the greatest common divisor of these
two last cannot exceed 13^ we must therefore try, if 13 will
divide 65, which is the case, and the quotient is 5 ; then 1 3 is
the greatest common divisbr sought.
We can make ourselverf certain of its possessing this property
by resuming the opera tio^ in an inverse order, as follows ;
As 13 divides 65 andi3, it will divide 143, which consists of
twice 65 added to 13; as it divides 65 and 143, it will divide
637, which consists of 4 times 143 adcled to 65 ; 13 then is the
common divisor of the two given numbers. It is also evident,
by the very mode of findmg it, that there can be no common
divisor greater than 13, since 13 must be divided by it.
iv
40
Jlrithmeiic*
143
4 130
65
2 65
5
13
It is convenient in practice, to place the successive divisions
one afC€r the other, and to dispose of the operation, as may be
seen in the following example ;
637 143 65 13
572
65
the quotients, 4, 2, 5, being separated from the other figures.
The reasoning, employed in the preceding example, may be
applied to any numbers, and thus conduct us to this general rule.
The greatest common divisor of two numbers will he founds by di-
viding the greater by the less ; then the less by the remainder of the
first droision ; then this remainder^ by the remainder of the second
division ; then this second remainder by the thirds or that of the
third division : and so on^ till we arrive at an exact quotient ; At
iast divisor will be tlie common divisor sought.
62. See two examples of the operation.
9024
7520
3760
2 300a
1504
2 1504
2
1504
752
00
752
752 then is the greatest common divisor of 9024 and 3760.
937
47
47
19 44
44
1 3
3
14 2
1
1
2
T|2
467
423
3
14
12
2
44
"21
1
By this last operation we see that the greatest common divi-
sor of 937 and 47, is 1 only, that is, these two numbers, properly
speaking, have no common divisor, since all whole numbers,
like them, are divisible by 1.
We may easily satisfy ourselves, that the rule of the preceding
article must necessarily lead to this result, whenever the given
numbers have no common divisor; for the remainders, each
being less than the correspondbg divisor, become less and less
every operation, and it is plain, that the division will continue
as long as there is a divisor greater tjian unity.
63. After these calculations, the fraction m and |||J^ can
Fraciioni* 41
be at once reduced to their most simple terms, by dividing the
terms dfthe first by their common divisor, 13, and the terms of
the second, by their common divisor, 752 ; we thus obtain \^
and j'y* As to the fraction, ^%-, it is altogether irreducible,
since its terms have no common divisor but unity.
64. It is not always necessary to find the greatest common
divisor of the given fraction ; there are, as has before been re-
marked, reductions, which present themselves without this pre-
paratory step.
Every number terminated by one of the figures, 0, 2, 4, 6, 8,
is necessarily divisible by 2 ; for in dividing any number by 2,
only 1 can remain from the tens ; the last partial division can
be performed on the numbers 0, 2, 4, 6, 8, if the tens leave no
remainder, and on the numbers 10, 12, 14, 16, 18, if. they do,
and all these numbers are divisible by 2.
The numbers divisible by 2 are called even numbers^ because
they can be divided into two equal parts.
Also, every number terminated on the right* by a cipher, or
by 5, is divisible by 5, for when the division of the tens by 5 has
been performed, the remainder, if there be one, must necessarily
be either 1, 2, 3, or 4, the remaining part of the operation will be
performed on the numbers 0, 5, 10, 15, 20, 25, 30, 35, 40, or 45,
all of which are divisible by 5.
The numbers, 10, 100, 1000, &c. expressed by unity followed
by a number of ciphers, can be resolved into 9 added to 1, 99
added to 1, 999 added to 1, and so on; and the numbers 9, 99,
999, &c. being divisible by 3, and by 9, it follows that, if num-
bers of the form 10, 100, 1000, &c. be divided by 3 or 9, the
remainder of the division will be 1.
Now every number which, like 20, 300, or 5000, is expressed
by a single significant figure, followed on the right by a number
of ciphers, can be resolved into several numbers expressed by
unity, followed on the right by a number of ciphers ; 20 is equal
lo 10 added to 10 ; 300, to 100 add'^d to 100 added to 100; 5000
to 1000 added to 1000 added to 1000 added to 1000 added to
.1000; and so with others. Hence it follows, that if 20, or 10
added to 10, be divided by 3 or 9, the remainder will fe 1 added
Arith. 6 #
4S jlrilhfwUc.
to 1, or 2 ; if 300, or 100 added to 100 added to 100, be divid-
ed by 3 or 9, the remainder will be 1 added to 1 added to4, or 3.
In general, if we resolve in the same manner a number ex-
pressed by one significant figure, followed, on the right, by a
number of ciphers, in ordei^ to divide it by 3 or 9 ; the remain-
der of this division will be equal to as many times 1, as there are
units in the significant figure, that, is, it will be equal to the signi-
ficant figure itself. Now any number being resolved into units,
tens, hundreds, &c. is formed by the union of several numbers
expressed by a single significant figure ; and, if each of these last
be divided by 3 or 9, the remainder will be equal to one of the
significant figures of the given number ; for instance, the division
of hundreds will give, for a remainder, the figure occupying the
place of hundreds ; that of tens, the figure occupying the place
of tens ; and so of the others. If then, the sum of all these re-
mainders be divisible by 3 or 9, the division of the given num-
ber by 3 or 9 can be performed exactly ; whence it follows, that*
if the sum of the figures, constituting any number, be divisible by
3 or 9, the number itself is divisible by 3 or 9.
Thus the numbers, 423, 4261^ 1534^% are divisible by 3, be-
cause the sum of the significant figures is 9 in the first, 12 in the
second, and 15 in the third.
Also, 621, 8280, 934218, are divisible by 9, because the sum
of the significant figures is 9 in the first, 1 8 in the second, and
27 in the third.
It must be observed, that every number divisible by 9 is also
divisible by 3, although every number divisible by 3 is not also
divisible by 9.
Observations might .be made on several other numbers analo-
gous to those just given on 2, 3, 5, and 9 ; but this would lead
me too far from the subject.
The numbers 1, 3, 5, 7, 11, 13, 17, &c. which can be divided
only by themselves, and by unity, are called prime numbers ; two
numbers, as 12 and 35, having, each of them, divisors, but
neither of them any one, that is common to it with the other, are
<:alled prime to each other.
Conseouently, the numerator and denominator of an irreduci^
ble fraclicn are prime to each other.
/
PracHonss 48
Exan^les for practice under Article 61.
What is the greatest common divisor of 34 and 36 ? Ans. 12.
What is the greatest common divisor of 35 and 100 ? Ans. 5.
What is the greatest common divisor of 312 and 504 ? Am. 24.
Examples for practice under articles 57, 58, and 60.
Reduce ^| to its most simple terms. Ans. |.
Reduce ^Vtt ^^ ^^^ ^^^^ simple terms. Ans. •}•
Reduce yVt ^^ ^^ ^^^^ simple terms. J^n^. ^*
Reduce \i^ to its most simple terms. Ans. |^.
Reduce 4^4 ^^ '^^ ^^^^ simple terms. Ans. ^.
Reduce ||f § to its most simple terms. Ans. \^.
65. After this digression we will resume the examination of
the table in article 55,
that we may deduce from it some new inferences.
We see at once, by an inspection of this table, that a fraction
can be multiplied in two ways, namely, by multiplying its nu-
merator, or dividing its denominator, and that it can also be
divided in two ways, namely, by dividing its numerator, or mul-
tiplying its denominator; hence it follows, that multiplication
alone, according as it is performed on the numerator or denomi-
nator, is sufScient for th^ multiplication and division of fractions
by whole numbers. Thus 7^, multiplied by 7 units, makes fj 3
^ divided by 3, makes ^^y.
Examples for practice.
Multiply I by 6. Ans. V* Divide | by 3. Ans. \.
Multiply ,\ by 4. Ans. |f . Divide •j\ by 6. Ans. ^j.
Multiply ^ by 6. Ans. |. Divide | by 10. Ans. ^j..
Multiply { by 30. Ans. *^*. Divide J by 8. Ans. J^.
Multiply ,V hy 5. Ans. J. Divide |^ by 4. Ans. L
Multiply :^ by 9. Ans. i. Divide 4f by 4. .^ns.^.
^ * AriAmeiie.
66. The doctrine of fractions enables us to generalize the
definition of multiplication given in article 21. When the multi-
plier is a whole number, it shows how many times the multipli-
« cand is to l^ repeated ; but the term multiplication, extended to
fractional expressions, does not always imply augmentation, as
in the case of whole numbers. To comprehend in one state-
ment every possible case, it may be said, that to mult^ly one
number by another 15, tofonn a number by means of the first j in tht
same manner as the second is formed^ by means of unit^. In real-
ity, when it is required to multiply by 2, by 3, &c. the product _
consists of twice, three tiroes, &c. the multiplicand, in the same
way as the multiplier consists of two, three, &c. units ; and to
multiply any number by a fraction, | for example, is to take the
fifth part of it, because the multiplier | being the fifth part of
unity, shows that the product ought to be the fifth part of the
multiplicand.*
Also, to multiply any number by ^ is to take out of this num-
ber or the multiplicand, a part, which shall be four fifths of it, or
equal to four times one fifth.
Hence it follows, that the object ifi multiplying by a fraction^
whatever may be tite multiplicand^ is^ to take out of tht multiplicand
a part, denoted by the multiplying fraction ; and that this opera-
tion is composed of two others, namely, a division and a multi-
plication, in which the divisor and multiplier are whole numbers.
Thus, for instance, to take ^ of any number, it is first neces-
sary to find the fifth part, by dividing the number by 5, and to
repeat this fifth part four times, by multiplying it by 4,
We see, in general, that the multiplicand fnust he divided by iht
denominator of the multiplying fraction, and the quotient be multi"
plied by its numerator*
The multiplier being less than unity, the product will be
smaller than the multiplicand, to which it would be only equal,
if the multiplier were 1.
f
* We are led to this statement, by a question which often presents
itseflf ; namely, where the price of any quantity of a thing is requir-
ed^ the price of the unity of the thing being known. The question
evidently remains the same^ whether the given quantity be greater
er less than this unity.
FraclionM. 45
67. If tbe multtplicand be a whole number divisible bj 5, for
instance, 35, the fifth part will be 7 ; this result, multiplied by 4,
ivill give 38 for the 4 of 35, or for the product of 35 by 4. If
the maUipIicand, always a whole number, be not exactly divisi-
ble by 5, as, for instance, if it were 32, the division by 5 will
give for a quotient 6| / this quotient repeated 4 times will give
24 »
T"
This result presents a fraction in which the numerator exceeds
the denominator, but this may be easily explained. The ex-
pression I , in reality denoting 8 parts, of which 5, taken to-
gether, make unity, it follows, that | is equivalent to unity added
to three fifths of unity, or 1| ; adding this part to the 24 units,
we have 25| for the value of | of 32.
88. It is evident, from the preceding example, that the frac-
tion I contains unity, o)f a whole one^ and |, and the reasoning,
which led to this conclusion, shows also, that every fractional
expression, of which the numerator exceeds the denominator,
contains one or more UYiits, or whole ones, and that these whole
ones may be extracted hy dividing the numerator by the denomina"
tor ; the quotient is the number of units contained in the fraction,
and tiie remainder, written as a fraction, is that which must accom-
pany the whole ones.
The expression '// , for instance, denoting 307 parts, of which
53 make unity, there are, in the quantity represented by this
expression, as many whole ones, as the number of times 53 is
contained in 307 ; if the division be performed, we shall obtain
5 for the quotient, and 42 for the remainder ; these 42 are fifty-
third parts of unity ; thus, instead of Vt 9 ^^y ^^ written 5|f.
•
Examples for practice.
Reduce the fraction f to its equivalent whole number.
Ans. 2.
Reduce } to its equivalent whole or mixed number. Ans. 3|^
Reduce y to its equivalent whole or mixed number.
« Ans. 3|.
Reduce V^^ ^^^ equivalent whole or mixed number.
Ans.. 347V
46 Jlriihmetk*
Reduce V to its equivalent whole or mixed number.
Ans. 12^.
Reduce V/ to its equivalent whole or mixed number. ,
Ans. 10/^.
69. The expression 6f |, in which the whole number is given,
being composed of two different parts, we have often occasion
to convert it into the original expression V? » which is called,
reducing a whole number to a fraction*
To do this, the whole number is to be multiplied by the derumii'
nator of the accompanying fraction^ the numerator to be added to the
product, and the denominator of the same fraction to be given to. the
sum*
In this case, the 6 whole ones must be converted into fifty-
thirds, which is done by multiplying 53 by 5, because each unit
must coiUain 63 parts ; the result will be ^/^ ; joining this part
with the second, J|, the answer will be y/«
Examples for practice*
,5.'^
Reduce t2| to a fraction. Ans* y.
Reduce 6| to a fraction. Ans. y«
Reduce 31 j\ to a fraction. ^ns, Vv •
Reduce 46yYy to a fraction. Ans. VVV*
70. We now proceed to the multiplication of one fraction by
another.
If, for instance, | wcreAo be multiplied by } ; according to
article 66, the operation would consist in dividing | by 5, and
multiplying the result by 4 ; according to the table in article 65,
the first operation is performed by mult)t>lying 3, the denomina-
tor of the multiplicand, by 5 ; and the second, by multiplying 2,
the numerator of the multiplicand, by 4 ; and the required pro-
duct is thus found to be j\.
It will be the same with every other example, and it must con-
sequently be concluded from what precedes, that to obtain the
product of two fractions, the two numerators must be multiplied^
one by the other, and under the product must be pimped the product
of the denominators.
Fractions. 47
Examplts,
Multiply \ by J. ^iw, ^. Multiply ^V by \. Ans* }.
Multiply I by f. vJrw. /^. Multiply f§ by ff. ^n^. |f.
^ Multiply f by |. ^rw. j%. Multiply \\ by H- ^^s- IH*
71. It may sometimes happen that two mixed numbers, or
whole numbers joined with fractions, are to be multiplied, one by
the other, as, for instance, 34 by 4|* The most simple mode of
obtaining the product is, to reduce the whole numbers to frac-
tions by the process in article 69 ; the two factors will then be
expressed by y and^y, and their product, by 4J* or I8|f,by
extracting the whole ones (68).
72. The name fractions of fractions is sometimes given to the
product of several fractions ; in this sense we say, | of |. This
expression denotes | of the quantity represented by J of the
original unit, and taken in its stead for unity. These two frac-
tions are reduced to one by multiplication (70), and the result,
f'j, expresses the value of the quantity required, with relation
to the original unit ; that is, | of the quantity represented by |
of unity is equivalent to j\ of unity. If it were required to take
} of this result, it would amount to taking j^ of f of |, and these
fractions, reduced to one, would give y^? ^^^ ^^^ value of the
quantity sought, with relation to the original unit.
73. The word contairif in its strict sense^ is not more proper in
-the different cases presented by division, than the word repeat in
those presented by multiplication ; for it cannot be said that tht
dividend contains the divisor, when it is less than the latter ; the
expression is generally used, but only by analogy and extension.
To generalize division, the dividend must be considered as hath
ing the same rehtion to the quotient^ that the divisor has to unity^
because the divisor and quotient are the two factors of the divi-
dend (36). This consideration is conformable to every case
that division can present. When, for instance, the divisor is 5,
the dividend is equal to 5 times the quotient, and, consequently,
this last is the fifth part of the dividend. If the divisor be a
fraction, ^ for instance, the dividend cannot be but half of the
quotient, or the latter must be the double of the former.
48 Ariihmetk*
«
The definition, just given, easily suggests the mode of pro-
ceeding, when the divisor is a fraction. Let us take, for ex-
ample, |. In this case the dividend ought to be only | of the
quotient ; but } being J of a, we shall have } of the quotient, by
taking i of the dividend, or dividing it by 4, Thus knowing |
of the quotient, we have only to take it 6 times, or multiply it
by 5, to obtain the quotient* In this operation the dividend is
divided by 4 and multiplied by 5, which is the same as taking
f of the dividend, pr multiplying it by f , which fraction is no
other than the divisor inverted.
This example shows, that, in general, to divide any number 6y
a fraction, it must be multiplied by the fraction inverted*
For instance, let it be required to divide 9 by f ; this will be
done by multiplying it by |^, and the quotient will be found to be
Y or 12. Also 13 divided by f will be the same as 13 multi-
plied by } or V- The required quotient will be 18}, by ex-
tracting the whole ones (68).
It is evident that, whenever the numerator of the divisor is
less than the denominator, the quotient will exceed the dividend,
because the divisor in that case, being less than unity, must be
contained in the dividend a greater number of times, than unity
is, which, taken for a divisor, always gives a quotient exactly
the same as the dividend. * .
-I
74. When the dividend is a fractimi, the operation amounts to
multiplying the dividend by the divisor inverted (70).
Let it be required to divide f by | ; according to the preced-
ing article, | must be multiplied by |, which gives ^J.
It is evident, that the above operation may be enunciated thus ;
To divide one fraction by another <, the numerator of the first must
he multiplied by the denominator of the second^ and the denominator
of the firsts by the numerator of the second*
If there be whole numbers joined to the given fractions, they
must be reduced to fractions, and the above rule applied to the
results.
V
Fraclian$* 49
Exan^lesM
Divide 9 by f. Ans. V- Divide 7 J by |. d3iw. V-
Divide 18 by |. ^ns. 15. Divide 2f by 3}. Ans. ||.
Divide f by ^. ^rw. T»f. Divide V by ^y. ^n^. 49.
Divide |f by ^V -^w*. ff. Divide f f by ff. j1/w. 1.
75. It is important to observe, that any division, whether it
can be performed in whole numbers or not, m^y be indicated by
a fractional expression; V, for instance, expresses evidently
the quotient of 36 by 3, as well as 12, for ^ being contained three
times in unity, y will be contained 3 times in 36 units, as the
quotient of 36 by 3 must be.
76. It may seem preposterous to treat of the multiplication and
division ojf fractions before having said any thing of the mapner
of adding and subtracting them ; but this order has been follow-
ed, because multiplication and division follow as the immediate
consequences of the remark given in the table of article 55, but
addition and subtraction require some previous preparation. It
is, besides, by no means surprising, that it should be more easy
.to multiply and divide fractions, than to add and subtract them,
since they are derived from division, which is so nearly related
to multiplication. There will be many opportunities, in what
follow^, of becoming convinced of this truth; that operations
to be performed on quantities are so much the more easy, as
they approach nearer to the origin of these quantities. We will
now proceed to the addition and subtraction of fractions.
77. When the fractions on which these operations are to be
performed have the same denominator, as they contain none but
parts of the same denomination, and consequently of the same
inagnitude or value, they can be added or subtracted in tjie
same manner as whole numbers, care bemg taken to mark, in the
result, the denomination of the parts, of which it is composed.
It is bdeed very plam, that -f^ smd ^j make /j-, as 3 quan-
tities and 3 quantities of the same kind make 5 of that kind,
whatever it may be.
Aritfu 7
50 ^ Ariihmeticm
Also, the difference between | and | is }, as the difference be-
tween 3 quantities and 8 quantities, of the same kind, is 5 of that
kind, whatever it may be. Hence it must be concluded, that, to
add or subtract fractions^ having the same denominator^ the sum or
difference of their numerators must he taken^ aud the common denonV'
tor written under the result.
78* When the given fractions have different denominators, it
is impossible to add together, or subtract, one from the other,
the parts of which they are composed, because these parts are
of different magnitudes ; but to obviate this difficulty, the frac*
tions are made to undergo a change, which brings them to parts
of the same magnitude, by giving them a common denominator.
For instance, let the fractions be f and f ; if each term of the
fii*st be multiplied by 5, the denominator of the second, the first
will be changed into |{ ; and if each term of the second be mul-
tiplied by 3, the denominator of the first, the second will be
changed into |f ; thus two new expressions will be formed, hav-
ing the same value as the given fractions (56).
This operation, necessary for comparing the respective mag-
nitudes of two fractions, consists simply in finding, to express
them, parts of an unit sufficiently small to be contained exactly
in each of those which form the given fractions. It is plain, in
the above example, that the fifteenth part of an unit will exactly
measure \ and \ of this unit, because | contains five 1 5^, and
i contains three 15****. The process, applied to the fractions |
and I, will admit of being applied to any others.
In general, to reduce any tv>o fractions to the same denominatoTj
ike §mo terms of each of them must be multiplied by th€ denominator
ef the otherm
(.
79. Any number effractions are reduced to a common denomtfux-
$or^ by multiplying the two terms of each by the product of the denomf^
inators of all the others ; for it is plain that the new denominators
are all the same, since each one is the product of all the original
denominators, and that the new fractions have the same value a*
the former ones, since nothing has been done except multiplying
each term of these by the same number (56).
Fractiom* 51
Examplesm
Reduce } and f to a common denominator. Jins. f }, f |.
Reduce j\ and ^ to a common denominator. An8. f }, ^|«
Reduce |, |, and f to a common denominator. Ans» f|, |f , ||.
Reduce i',^, |, 4? and {• to a common denominator.
^ns^ tVtV, if**, ill*, IH*.
The preceding rule conducts us, in s^ll cases, to the proposed
end; but when the denominators of the fractions in question are
not prime to each other, there is a common denominator more
simple than that which is thus obtained, and which may be
shown to result from considerations analogous to those given in
the preceding articles. If, for instance, the fractions were |, |,
f , }, as nothing more is required, for reducing them to a com-
mon denominator, than to divide unity into parts, which shall be
exactly contained in those of which these fractions consist, it will
be sufficient to find the smallest number, which can be exactly
divided by each of their denominators,' 3, 4, 6, 8 ; and this will
be discovered by trying to divide the multiples of 3 by 4, 6, 8 ;
which does not succeed until we come to 24, when we have only
to change the given fractions into 24^ of an unit.
To perform this operation we must ascertain successively how
many times the denominators, 3, 4, 6, and 8, are contained in
S4, and the quotients will be the numbers, by which each term
of the respective fractions must be multiplied, to be reduced to
the common denominator, 24. ' It will thus be found, that each
term of | must be multiplied by 8, each term of J by 6, each
term of f by 4, and each term of | by 3 ; the fractions will then
1?ecome if, if, |J, fj.
Algebra will furnish the means of facilitating the application
of this process.
80. By reducing fractions to the same denominator, they may
be added and subtracted as in article 77.
81. When there are at the same time both whole numbers and
fractions, the whole numbers, if they stand alone, must be con-
Verted into fractions of the same denomination as those which
52 AridmuHc.
are to be added to them, or subtracted from them ; and if the
whole numbers are accompanied with fractions, they must be
reduced to the same denominator with these fractions*
It is thus, that the additi6n of four units and ^ changes itself
into the addition of y and |, and gives for the result y* •
To add 34 to 5|-, the whole numbers must be reduced to frac-»
tions, of the same denomination as those which accompany them,
which reduction gives Y and y ; with these results the sum is
found to be V/) ^^ ^H* ^^y lastly, | were to be subtracted from
3}, the operation would be reduced to taking } from y, and the.
remainder would be f }•
Examples in addition of fractions*
Add I to f. Ans. |^, or 1.
Add 4 to H- *^^* 7i«
Add I to 4* Ans. \i.
Add f , }, and | together. Ans. Sy'j.
Add ^^ 4f, and 5| together. Ans. 13tV
^^^ h H> ^"^ ^1 together. Ans. 8|.
Examples in 9yJJ>traciion of fractions^
From I take \. Ans. ^. From 5} take ^» Am. 2|.
From i take f • Ans. Z^. From 8| take 4{. Ans, 4^^.
F»)m If take y\. Ans. {. . From 3| take ^f. Ans. |f •
82. The rule given for the reduction of fractions to a common
denominator supposes, that a product resulting from the succes-
sive multiplications of several numbers into each other, does not
vary, in whatever order these multiplications may be performed ;
this truth, though almost always considered as self-evident, needs
to be proved.
We shall begin with showing, that to multiply one number by
the product of two others, is the same thing as to multiply it at
first by one of them, and then to multiply that product by the
other. For instance, instead of multiplying 3 by 35, the pro-
duct of 7 and 5, it will be the same thing if we multiply 3 by 5,
and then that product by 7. The proposition will be evident, if.
Fractwri^. &S
instead of 3, #e tdke an unit; for }, muUipfied by 5, gives 5,
and tlie product of 5 by 7 is 35, as well as the product of 1 by
35 ; but 3, or any other number, being only an .assemblage of
several units, the same property will belong to it, as to each of
the units of which it consists ; that is, the product of 3 by 5 and
by 7, obtained in either way, being the triple of the result given
by unity, when multiplied by 5 and 7, must necessarily be the
same. It may be proved in the same manner, that were it re-
quired to muhiply 3 by the product of 5, 7, and 9, it would
consist in multiplying 3 by 5, then this product by 7, and the
result by 9, and so on, whatever might be the number of factors*
To represent in a shorter manner several successive multipli-
cations, as of the numbers 3, 5, and 7, into each other, we 3haU
write 3 by 5 by 7,
This being laid down, in the product 3 by 5, the order of the
factors, 3 and 5 (27), may be changed, and the same product
obtained. Hence it directly follows, that 5 by 3 by 7 is tho^
same as 3 by 5 by 7.
The order of the factors 3 and 7, in the produot 5 by 3 by 7,
may also be changed, because this product is equivalent to 5.^
multiplied by the product of the numbers 3 and 7 ; thus we have
in the expression 5 by 7 by 3, the same product as the preceding*
By bringing together the three arrangements,
3 by 5 by 7
5 by 3 by 7
5 by 7 by 3,
we see that the factor 3 is found successively, the first, the
second, and the third, and that the same may take place with
respect to either of the others. From this example, in which the
particular value of each number has not been considered, it must
be evident, that a product of three factors does not ^ry, what-
ever may be the order in which they are multiplied.
If the question were concerning the product of four factors,
such as 3 by 5 by 7 by 9, we might, according to what has been
said, arrange, as we pleased, the three first or the three last, and
thus D\^e any one of the factors pass through all the places*
Considering then one of the new arrangements, for instance thiS|
54 Arithmetic.
5 by 7 bj 3 by 9, we might invert the order of the two last fac-
tors, which would give 5 by 7 by 9 by 3, and would put 3 in the
last place. This reasoning may be extended without difficulty
io any number of factors whatever.
Decimal Fractions.
83. Although we can, by the preceding rules, apply to frac-
tions, in all cases, the four fundamental operations of arithmetic,
yet it must have been long since perceived, that, if the different
subdivisions of a unit, employed for measuring quantities smaller
than this unit, had been subjected to a common law of decrease,
the calculus of fractions, would have been much more conven-
ient, on account of the facility with which we might convert one
into another. By making this law of decrease conform to the
basis of our system of numeration. We have given to the calculus
the greatest degree of simplicity, of which it is capable.
We have seen in article 5, that each of the collections of units
contained in a number, is composed of ten units of tbe preceding
order, as the ten consists of simple units ; but there is nothing
to prevent our regarding this simple unit, as containing ten parts,
of which each one shall be a tenth ; the tenth as containing ten
parts, of which each one shall be a hundredth of unity, the hun- '
dredth as containing ten parts, of which each one shall be a
thousandth of unity, and so on.
Proceeding thus, we may form quantities as small as we please,
by means of which it will be possible to measure any quantities,
however minute. These fractions, which are called decimals,
Because they are composed of parts of unity, that become con-
tinually ten times smaller, as they depart further from unity,
may be converted, one into the other, in the same manner as
tens, hundreds, thousands^ &c. are converted into units ; thus,
the unit being equivalent to 10 tenths,
the tenth 10 hundredths,
the hundredth 10 thousandths,
it follows, that the tenth is equivalent to 10 times 10 thousandths,
or 100 thousandths.
Decimal Fractions* 55
For instance, 2 tenths, 3 hundredths, and 4 thousandths will
be equivalent to 234 thousandths, as 2 hundreds, 3 tens, and 4
units make 234 units ; and what is here said may be applied
universally, since the subordination of the parts of unity is like
that of the different orders of unils*
84. According to this remark, we can, by means' of figifres^
write decimal fractions in the same manner as whole numbers,
since by the nature of our numeration, which makes the value of
a figure, placed on the right of another, ten times smaller, tenths
naturally take their place on the right of units, then hundredths
on the right of tenths, and so on : but, that the figures express-
ing decimal parts may not be confounded with those expressing
whole units, a comma! is placed on the right of units. To ex-
press, for instance, 34 units and 27 hundredths, wc write 34,27.
If there be no units, their place is supplied by a cipher, and the
same is done for all the decimal parts, which may be wanting
between those enunciated in the given number.
Thus 19 hundredths are written 0,19,
304 thousandths 0,304,
3 thousandths 0,003.
85. If the expressions for the above decimal fractions be com-*
pared with the following, j'/y? tVA? tt^tt? drawn from the gen-
eral manner of. representing a fraction, it will be seen, that to
represent in an entire form a decimal fraction,, written as q vulgar
fraction^ the numerator of the fraction must be taken as it is^ and
placed after the comma in such a manner^ that it may have as many
figures as there are ciphers after the unit in the denominator.
Reciprocally, to reduce a decimal fraction, given in the form of
a whole number j to that of a vulgar fraction, the figures that it con--
tains, must receioe,for a denominator, an unit followed by as many
ciphers, as there are figures after the comma.
t Id English books on mathematics, and in those that have been
written in the United States, decimals are usually denoted by a
point, thus 0.19 ; but the comma is on the whole in the most general
use ; it is accordingly adopted in this and the subsequent treatises
published at Cambridge.
4
d6 Arithmetic.
Thus the fractions, 0,S6, 0,036, are changed into iV^ and
86. An expression, in figures j of numbers containir^ decimal
parls^ is read by enunciating^ fi^^t^ the figures placed on the left of
the pointy then those on the rights adding to the last figure of the
latter the denomination oftheparts^ which it represents.
The number 36,736 is read 26 and 736 thousandths ;
the number 0,0673 is read 673 ten-thousandths,
and 0,0000673 is read 673 ten-millionths.
87. As decimal figures take their value entirely from their
position relative to the comma, it is of no consequence whether
we write or omit any number of ciphers on their right. For
instance, 0,5 is the same as 0,50; and 0,784 is the same as
€,78400; for, in the first instance, the number, which expresses
the decimal fraction, becomes by the addition of a ten times
greater, but the parts become hundredths, and consequently on
this account are ten times less than before ; in the secondT in-
stance, the number, which expresses the fraction, becomes a
hundred times greater than before, but the parts become hun-
clred-thousandths, and, consequently, arc a hundred times smaller
than before. This transformation, then, becomes the same as
that which takes place with respect to a vulgar fraction, when
/each of its terms is multiplied by the same number; and if the
ciphers be suppressed, it is the same as dividing them by the
isame number.
88. The addition of decimal fractions and numbers aocompa,-
nying them, needs no other rule than that given for the whole
numbers, since the decimal parts are made up one from the other,
ascending from right to left, in the same manner as whole units.
For instance, let there be the number^ 0,56, 0,003, 0,958.;
4]isposing them as follows,
0,56
t, 0,003
0,958
Sum 1,521
we find, by the rule of article 12, that their sum is 1,521
Decimal Fractions SI
Again, let there be the numbers 19,35, 0,3, 48,5, and 1 10,09,
which contain also whole units ; they will be disposed thus \
19,35
0,3
48,5 •
110,02
Sum 178,17
and their sum will be 178,17.
In general, the addition of decimal numbers is perfimned like
that of whole numbers^ care being taken to place the comma in th^
ntm, directly under the commas in the numbers to be added.
Examples for practice.
Add 4,003, 54,9, 3,21, 6,7203. Ans. 68,8333.
Add 409,903, 107,7842, 6,1043, 10,2074. ^/w. 533,9989.
Add 427, 603,04, 210,15, 3,364, ,021. jins. 1243,575.
89. The rules prescribed for the subtraction of whole num-
bers apply also, as will be seen, to decimals. For instance, let
0,3697 be taken from 0,62 ; it must first be observed, that the
second number, which contains only hundredths, while the other
contains ten-thousa^ndths, can be converted into ten-thousandths
by placing two ciphers on its right (87), which changes it into
0,6200.
The operation will then be arranged thus i
0,6200
0,3697
Difference 0,2503
and, accordmg to the rule of article 17, the difference will be
0,2503.
Again, let 7,364 be taken from 9,1457; the operation being
disposed thus ;
9,1457
7,3640
Difference 1,7817
Arith. 8
dft Arithmetic*
the above difference is found. It would have been just as well
if no cipher had been placed at the end of the number to be sub<
traded, provided ils different figures had been placed under the
corresponding orders of units or parts, in the upper line.
In general, the subtraction of decimal numbers is performed like
that of whole numbers^ prorcided that the number of decimal figures j
in the two given numbers^ be made alike^ by writing on the right of
that which has the leasts as many ciphers as are necessary ; and that
the comma in the difference be put directly under those qf the given:
numbers*
Examples for practice.
From 304,567 take 158,632. Jtns. 145,935.
From 215,003 take 1,1034. Ans. 213,8996.
From V / ; 'o ^^^^ ,9993. Ans. 0,0007.
From 68^8333 take ,00042. Ans. 68,83288.
The methods of proving addition and subtraction of decimals
are the same as those for the addition and subtraction of whole
numbers.
90. As the comma separates the collections of entire units
from the decimal parts, by altering its place, we necessarily
change the value of the whole. By moving it towaixis the right,
figtires, which were contained in the fractional part, are mad^ to
pass into that of whole numbers, and consequently the value of
the given number is increased. On the contrary, by moving the
comma towards the left, figures which were contained in the part
of whole numbers, are made to pass into that of fractions, and
consequently the value of the given number is diminished.
The first change makes the given number, ten, a hundred, a
thousand, &,c. times greater than before, according as the comma
is removed one, two, three, &c. places towards the right, because
for each place that the comma is thus removed, all the figures
advance with respect to this comma one place towards the left,
and consequently assume a value ten times greater than they had
before.
If, for example, in the number 134,28, the point bp placed
Decimal Fractions. 39
between the 3 and the 8, we shall have 1342,8, the hundreds
will have become thousands, the tens hundreds, the units tens,
the tenths units, and the hundredths tenths. Every part of the
number having thus become ten times greater, the result is the
same as if it had been multiplied by ten.
The second change makes the given number ten, a hundred, a
thousand, &c« times smaller than it was before, according as the
comma is removed one, two, three, &;c. places towards the Ic;ft ;
because for each place that the comma is thus removed, all the
figures recede, with respect to this comma, one place further to
the right, and consequently have a value ten times less than they
had before.
If, in the number 134,28, the point be placed between the 3
and 4, we shall have 13,428 ; the hundreds will become tens,
the tens units, the units tenths, the tenths hundredths, and the
hundredths thousandths ; every part of the number having thus
become ten times smaller, the result is the same as if a tenth
part of it had been taken, or as if it had been divided by ten.*
91. From what has been said, it will be easy to perceive the
advantage, which decimal fractions have over vulgar fractions ;
all the multiplications and divisions, which are performed by
the denominator of the latter, are performed with respect to the
former, by the addition or suppression of a number of ciphers,
or by simply changing the place of the comma. By adapting
these modifications to the theory of vulgar fractions, we thence
immediately deduce that of decimals, and the manner of perform-
ing the multiplication and division of them ; but we can also
arrive at this theory directly by the following considerations.
Let us first suppose only the multiplicand to have decimal
figures. If the comma be taken away, it will become ten, a
hundred, a thousand, &;c. times greater, according to the number
of decimal figures; and in this case the product given by multi-
plication will be a like number of times greater than the one
required; the latter will then be obtained by dividing the former
by ten, a hundred, a thousand, &;c. which may be done by sep-
arating on the right (90) as many decimal figures, as there are
in the multiplicand.
60 Arithmetic.
If, for instance, 34,137 were to be multiplied by 9, we must
first find the product of 34137 by 9, which will be 307233 ; and
since taking away the comma renders the multiplicand a thou-
sand times greater, we must divide this product by a thousand,
or separate by a comma its three last figures on the right ; we
shall thus have 307,233.
In general, to multiply^ by a whole nnmber, a number accompa"
nied by decimals^ the comma must be taken away from trie mulbpli'
cand^ and as many figures separated for decimals, on the right of
the product, as are contained in the multiplicand.
Examples for practice.
Multiply 231,415 by 8. Ans. 1851 320.
Multiply 32,1509 by 15, Ans. 482.2635.
Multiply ,840 by 840. Ans. 705,600.
Multiply 1,236 by 13. Ans. 16,068.
92. When the multiplier contains decimal figures, by sup-
pressing the comma, it is made ten, a hundred, a thousand, &c.
times greater according to the number of decimal figures. If
used in this state, it will evidently give a product, ten, a hun-
dred, a thousand, &c. times greater than that which is req?jircd,
and consequently the true product will be obtained by dividing
by one of these numbers, that is, by separating, on the right of
it, as many decimal figures as there are in the multiplier, or by
removing the comma a like number of places towards the left
(90), in case it previously existed m the product on account of
decimals in the multiplican(J. For instance, let 172,84 be multi-
plied by 36,003 ; taking away the comma in the multiplier only,
we shall have, according to the preceding article, the product
6222758,52; but, the multiplier being rendered a tljousand limes
too great, we must divide this product by a thousand, or remove
the comma three places towards the left, and the required pro-
duct will then be 6222.75852, in which there must necessarily
be as many decimal figures as there are in both multiplicand and
multiplier.
In general, to multiply, one by the other, two numbers accompa'
nied by decimals, the comma must he taken away from both, and as
Decimal Fractiotis* 61
many Jigures sq^arated for decimals^ on the right of the product^ as
there are in both thefactors* ,
In some cases it is necessary to put one or more ciphers on
the left of the product, to give the number of decimal figures re-
quired by the above rule. If, for example, 0,624 be multiplied
by 0,003 ; in forming at first the product of 624 by 3, we shall
have the number 1872, containing but 4 figures, and as 6 figures
iDUSt be separated for decimals, it cannot be done except by
placing on the left three ciphers, one of which must occupy the
place of units, which wilt make 0,001872.
Examples for practice.
Multiply 223,86 by 2,500. Ans. 559,65000.
Multiply 35,640 by 26,18. Ans. 933,05520.
Multiply 8,4960 by 2,618. Ans. 22,2425ti80.
Multiply ,5236 by ,2808. jJn*. 0,14702688.
Multiply , 1 1 785 by ,27. Ans. 0,03 16195.
93. It is evident (36), that the quotient of two numbers does
not depend on the absolute magnitude of their units, provided
that this be the same in ea^h ; if then, it be required to divide
451,49 by 13, we should observe that the former amounts to
45149 hundredths, and the latter to 1300 hundredths, and that
these last numbers ought to give the same quotient, as if they
expressed units. We shall thus be led to suppress the point in
the first number, and to put two ciphers at the end of the second,
and then we shall only have to divide 45149 by 1300, the quo-
tient of which division will' be 34tVtV
Hence we conclude, that to divide^ by a whole number^ a num-
ber accompanied by decimal figures^ the comma in the dividend must
be taken away^ and as many ciphers placed at the end of the divisor^
as the dividend contains decimal fgures^ and no alteration in the
quotient will be necessary.
94. When both dividend and divisor are accompanied by de-
cimal figures, we must, before taking away the comma, reduce
them to decimals of the same order, by placing at the end of that
number, which has the fewest decimal figures, as many ciphers
6 2 Ariihnutk.
as will make it terminate at the same place 6f decimals as the
other, because then the suppression of the comma rendei*s both
the same number of times greater.
For instance, let 315,432 be divided by 23,4, this last must be
changed into 23,400, and then 315432 must be divided by
23400; the quotient virill be 13mj§.
Thus, to divide^ one by the othtr^ two numbers accompanied by
decimal figures^ the number of decimal figures in the divisor and
dividend must be made equal, by annexing to the one that has the
least, as many ciphers as are necessary ; the point must then be sup^
pressed in each, and the quotient will require no alteration.
95. As we have recourse to decimals only to avoid the neces-
sity of employing vulgar fractions, it is natural to make use of
decimals for approximating quotients that cannot be obtained
exactly, which is done by converting the remainder into tenths,
hundredths, thousandths, &,c. so that it may contain the divisor ;
as may be seen in the following example ;
45149
3900
1300
34,73
-
6149
4 ■
5200
Remainder
949
tenths
9490
9100
hundredths
3900
390(
3
When we come to the remainder 949, we annex a cipher in
order to multiply it by ten, or to convert it into tenths ; thus
forming a new partial dividend, which contains 9490 tenths and
gives for a quotient 7 tenths, which we put on the right of the
units, after a comma. There still remains 390 tenths, which we
reduce to hundredths by ihe addition of another cipher, and
form a second dividend, which contains 3900 hundredths, and
Decimal Fractions.
63
gives a quotient, 3 hundredths, which we place after the tenths.
Here the operation terminates, and we have for the exact result
34,73 hundredths. If a third remainder had been left, we might
have continued the operation, by converting this remainder into
thousandths, and so on, in the same manner, until we came to an
exact quotient, or to a remainder composed of parts so small,
that we might have considered them of no importance.
It is evident, that we must Sflwajs put a comma, as in the
above example, after the whole units in the quotient, to distin-
guish them from the decimal figures, the number ^f which must
be equal to that of the ciphers successively written after the
remainders.*
Examples for practice.
Divide 6346,925
by 54,23.
Ans. 117,018 tc.
Divide 5673^1
by 23,0.
jins. 246,6 6«|&:c.
Divide 84329907
by 627,1.
Ans. 134476,01 &c.
Divide 27845,96
by 9,S732.
Ans. 2820,3581 Slc.
Divide 200,5
by 331.
Ans. 0,0867 &c.
Divide 10,0
by 563,0.
Ans. 0,00177 &c.
Divide 513,2.
by 0,057.
Ans. 9003,50 &c.
Divide 7,26406
by 957
Ans. 0,00758
Divide 0,00078759
by 0,525.
Ans. 0,00150 &c.
Divide 14
by 365.
Ans. 0,038356 &c.
96. The numerator of a fraction, being converted into decimal
parts, can be divided by the denominator as in the preceding
examples, and by this means the fraction will be converted into
decimals. Let the fraction, for example, be |, the operation is
performed thus ;
"^ The problem above performed with respect to decimals, is only
a particular case of the following more general one ; To find the
value of the quotient of a division^ in fractions of a given denomina-
tion ; to do this we convert the dividend into a fraction of the same
denomination by multiplying it by the given denominator. Thus, in
order to find in fifteenths the value of the quotient of 7 by 3, we
should multiply 7 by 15, and divide the product, 105, by 3, which
gives thirty-five fifteenths, or f^|, for the quotient required.
<4
Arithmetic
1
8
10
0,125
8
20
16
40
40
Again, let the fraction be ^1^ ; the numerator must be con-
verted into thousandths before the division can begin.
797
4000
3985
0,005018 &c.
1500
797
7030
6376
654
Examples for practice.
\
Reduce J to a decimal fraction.
Reduce | to a decimal fraction.
Reduce ^V to a decimal fraction.
Reduce rf 7 to a decimal fraction.
Reduce f to a decimal fraction.
Am. 0,75.
Ans, 0,5.
Ans. 0,0714285 &;c.
Ans. 0,05.
Ans. 0,333 &LC.
* It may also be proposed to convert a given fraction into a frac-
tion of another denomioation, but smaller than the first, for instance,
|- into seventeenths, which will be done bj multiplying 3 by 17 and
dividing the product by 4. In this manner we find ^^ seventeenths,
or j-^ and f of a seventeenth ; but ^ of ^ is equivalent to ^. The
result then, j^^, is equal to f , wanting ^.
This operation and that of the preceding note depend on the same
priqcipie, as the corresponding operation for decimal fractions.
Decimal Fractions. 65
97. However far we may continue the second division, exhib-
ited above, we shall never obtain an exact quotient, because the
fraction if^if cannot, like |, be exactly expressed by decimals.
The diSerence in the two cases arises from this, that the de-
nominator of a fraction, which does not divide its niunerator,
canndt give an exact quotient, except it will divide one of the
numbers 10, 100, 1000, &,c. by which its numerator is succes-
sively multiplied ; because it is a principle, which will be found
demonstrated in Algebra, that no number will divide a product
except its factors will divide those of the product ;. now the num-
bers 10, 100, 1000, &c. being all formed from 10, the factors
of which are 2 and 5, they cannot be divided except by numbers
formed from these same factors; 8 is among these, being the
product of 2 by 2 by 2.
Fractions, the value of which cannot be exactly found by de-
cimals, present in their approximate expression, when k has
been carried suflSciently far, a character which serves to denote
them ; this is the periodical return of the same figures.
If we convert the fraction ^f into decimals, we shall find it
0,324324 , and the figures 3, 2, 4, will always return in
the same order, without the operation ever coming to an end.
Indeed, as there can be no remainder, in these successive
divisions, except one of the scries of whole numbers, 1, 2, 3, &,c.
up to the divisor, it necessarily happens, that, when the number
of divisions exceeds that of this series, we must fall again upon
some one of the preceding remainders, and consequently the
partial dividends will return in the same order. In the above
example three divisions are sufficient to cause the return of the
same figures; but six are necessary for the fraction |, because
in this case we find, for remainders, the six numbers which are
below 7, and the result is 0,1428571 . . • The fraction ^ leads
only td 0,3333
98. The fractions, which have for a denominator any number
of 9s, have no significant figure in their periods except 1 ;
i gives 0,11111 . .,
^\ 0,010101
vir 0,001001001
Ariih. 9
66 Arithmetic^
and so with the others, because each partial division of the num-
bers 10, 100, 1000, &c. always leaves unity for the remainder.
Availing ourselves of this remark, we pass easily from a peri-
odical decimal, to the vulgar fraction from which it is derived.
We see, for example, that 0,33333 amounts to the same
as 0,1 1 1 1 1 . . . • # multiplied by 3, and as this last decimal is
the development of |, or ^ reduced to a decimal, we conclude,
that the former is the development of | multiplied by 3, or f , or
lastly, 1.
When the period of the fraction under consideration consists
of two figures, we compare it with the development of ^V, and
with that of ^^7, when the period contains three figures, and
so on.^
If we had, for exampIeV 0,324324, it is plain that this fraction
may be formed by multiplying 0,001001 by 324 ; if we
multiply then ^^7, of which 001001 is the development^
by 324, we obtain ||f , and dividing each term of this result by
27, we come back again to the fraction |f .
In general, the vulgar fraction^ from which a decimal fraction
arises^ is formed by writings as a derwminator^ under the number^
which expresses one period^ as many 9*, as there are Jlgures in the
period.
If the period of the fraction does not commence with the first
decimal figure, Wvcan for a moment change the place of the point,
and put it immediately before the first figure of the period and
beginning with this figure, find the value of the fraction, as if
those figures on the left were units ; nothing then will be neces-
sary except to divide the result by 10, 100, 1000, &c. according
to the number of places the point was moved towards the right*
For instance, the fraction 0,324,141 . . . . , is first to be written
32,4 1 4 1 . . . . ; tfie part 0,4 141.... being equivalent to ^|, we shall
have 32|J, which is to be divided by 100, because the point was
moved two places towards the left ; it will consequently become
yYir 3nd ^Jiy, or by reducing the two parts'to the same denomi-
nator, and adding them, |||}, a fraction Vhich will reproduce
the given expression.
Decimal Fractions. 67
»
Example for practict,*
Reduce 0,18 id the form of a vulgar fraction. Ans. A-.
* •• . f
Reduce 0,72 to the form of a vulgar fraction* Ans. tV*
Reduce QfiS to the form of a vulgar fraction. Ans. |.
• • • '
Reduce 0,2418 to the form of a vulgar fraction. Ans. |}|}.
Reduce 0,375463 to the form of a vulgar fraction.
Reduce 0,916 to the form of a vulgar fraction* Ans. |^.
T^o form a correct idea of the nature of these fractions it is
sufficient to consider the fraction 0,999 In trying to discover
its original value we find that it answers to 9 divided by 9, that is,
to unity ; nevertheless, at whatever number of figures we stop in
its expression, it will never make an unit. If we stop at the first
figure, it wants ^V of an unit; if at the second, it wants ji-g ; if
at the third, it wants y^Vir? ^^^ ^^ ^" ^ ^^ ^^^^ ^^ ^^^ arrive as
near to unity as we please, but can never reach it. Unity then
in this case is nothing but a limits to which 0,999 .••... con-
iinually approaches the nearer the more figures it has.f
99. The preceding part of this work contains all the rules
absolutely essential to the arithmetic of abstract num^rs, but to
apply them to the uses of society it is necessary to know the
•different kinds of units, which are used to compare together, or
ascertain the value of quantities, under whatever form they may
present themselves. These units, which are the measures in use,
have varied with times and places, and their connexion has been
formed only by degrees, accordingly as necessity and the pro-
gress of the arts and sciences have required greater exactness in
the valuation of substances, and the construction of instruments.
* In these examples, the better to distinguish the period, a point
is placed over it, if it be a single figure, and over the first and last
figure, if it consist of more than one.
t See note on continued fractions at the end of this treatise.
68 Arithmetic.
Tables of Coin, Weight, and Measure.
Denominations of Federal money, as established by act of
Congress, April 2, 1792.1
marked m
c
d
$
E
Mills .
10 Mills make
1 Cent
10 Cents
1 Dime
10 Dimes
1 Dollar
10 Dollars
1 Eagle
t The act aboTe referred to provides, that the money of account
of the United States shall be expressed in dollars or units, and dimes
or tenths of a dollar, cents or hundredths of a dollar, and mills or
thousandths of a dollar.
The coins of gold, silver, and copper, shall be of the following
denominations, namely ;
1. Gold — eagle^ half^eagle^ quarter- eagle.
2. Silver — dollar, half-dollar, quarter-dollar, dime, half-dime,
3. Copper — ctnt^ half-cent.
The standard for all gold coin shall be eleven parts of pure gold
and one part alloy in twelve parts of the coin. The alloy is to be
of silver and copper, the silver not exceeding one half in the alloy.
The eagle shall contain 247^ grains of pure gold, or 270 grains of
standard gold. In the other gold coins the weight of pure or stand-
ard gold shall be in proportion to their values.
The standard for all silver coins shall be 1485 parts of pure silver
to 179 parts of alloy; and the alloy shall be pure copper.
The dollar shall contain 371^ grains of pure silver, or 416 grains
of standard silver. In the other silver coins the weight shall be in
proportion to their respective values.
The copper coins shall be pure copper. The cent shall contain
11 pennyweights of copper; and the half-cent half this weight of
copper.
In practical treatises on arithmetic, may be found rules for reduc-
ing the Federal Coin, the currencies of the several United States,
and those of foreign countries, each to the par of all the others. It
may be sufficient here to observe respecting the currencies of the
several states, that a dollar is considered as 6^. in New England and
Tables ofCoin^ Weighty and Measurt*
69
English Money.
2 Farthings
= 1
Halfpenny ^
qrs d
4 Farthings
= 1
Penny <
d
4= 1
s
12 Pence
= 1
Shilling
s
48= 12
= 1 £
20 Shillings
= 1
Pound £
960 = 240
= 20= 1
Pence Table.
Shillinf>
s Table.
d
s
d
s
d
20 is
1
8
1 is
12
30
2
6
2
24
40
3
4
3
36
50
4
2
. 4
48
60
5
5
60
70
5
10
■
6
72
80
6
8
7
84
90
7
6
8
96
100
8
4
9
108
110
9
2
f
10
120
120
10
11
132
Troyh
hi
■ght.
Grains
•
raar
ke
dgr
^r
dwt
24 Grains make
1 Pennyweig
;ht
. dwi
24 =
1 oz
20 Pennyweights
1 Ounce
oz
480 =
: 20= 1 ft
12 Ounces
1 Pound
lb
5760 =
240 = 12 = 1
By this weight are weighed gold, silver, and jewels.
Apothecaries'^ Weight.
Grains - marked gr.
20 Grains make 1 Scruple sc or ^
3 Scruples
1 Dram dr or 5
8 Drams
1 Ounce oz or |
12 Ounces
1 Pound Z6orlb
^*-
sc
20 =
1 dr
60 =
3=1 oz
480 =
24= 8= 1 lb
5760 =
268 = 96 = 12 = 1
Virginia ; 8j. in New York and North Carolina ; 7«. 6c?. in New
Jersey, Pennsylvania, Delaware, and Maryland ; and 4^. M. in South
Carolina and Georgia ; the denomination of shilling varying its value
accordingly.
,ii
70 Arthnut'c.
This is the same as Troy weighty only having some different
divisions. Apothecaries make use of this weight in compound-
ing their medicines ; but they buy and sell their drugs by Avoir-
dupois weight.
Avoirdupois tVtight,
Drams -
marked dr
16 Drams make 1 Ounce
oz
16 Ounces 1 Pound
lb
28 Pounds 1 Quarter
V
4 Quarters 1 Hundred
weight cvot
20 Hundred weight 1 Ton
ton
dr 02
16= 1 ^ Ih
256 = . 16 = 1 qr
7168= 448= 28= 1
cwi
28672= 1792= 1!2= 4 =
1 ton,
573440 = 35840 = 2240 = 80 = 20 = 1
By this weight are weighed all things of a coarse or drossy
nature, as corn, bread, butter, cheese, flesh, grocery wares, and
some liquids ; also all metals, except silver and gold.
^ 02 dwt gr
J^otCj that 1/6 Avoirdupois =14 11 15^ Troy.
loz = 18 5i
Idr = 1 3^
Hence it appears that the pound Avoirdupois contains 6999^
grains, and the pound Troy 5760; the former of which aug-
mented by half a grain becomes 7000, and its ratio to the latter
is therefore very nearly as 700 to 576, that is, as 175 to 144;
consequently 144 pounds Avoirdupois are very nearly equal to
175 pounds Troy ; and hence we infer that the ounce Avoirdu-
pois is to the ounce Troy as 175 to 192.
Tables ofCoin^ Weighty and Measure* 71
Lang Measure,
3 BarlejcoTDS make 1 Inch In
12 Inches i Foot Ft
3 Feet 1 Yard Yd
6 Feet 1 Fathom Fih
5 Yards and a half 1 Pole or Rod PI
40 Poles 1 Furlong Fur
8 Furlongs 1 Mile Mile
3 Miles 1 League Lea
69} Miles nearly 1 Degree Deg or ^
In Ft
12= 1 Yd
36= 3= 1 PI
198= 16^= 6}= 1 Fur
7920 = 660 = 220 = 40 = 1 Mile
63260 = 5280 = 1760 = 320 = 8 = 1
Cloth Measure.
2 Inches and a quarter make 1 Nail Jft
4 Nails 1 Quarter of a Yard Qr
3 Quarters 1 Ell Flemish E F
4 Quarters 1 Yard Yd
5 Quarters 1 Ell English fi E E
A Quarters 1} Inch 1 Ell Scotch t^\C\ E S
Square Measure.
144 Square Inches make 1 Sq. Foot Ft
9 Square Feet 1 Sq, Yard Yd
30J Square Yards 1 Sq, Pole Pole
40 Square Poles 1 Rood Rd
4 Roods 1 Acre Jlcr
Sq Inc Sq Feet
144 = 1 Sq Yd
1296= 9 = 1 SqPl
89204= 272}= 30}= 1 Rd
1568160 = 10890 =1210 = 40 ±= 1 Aer
6272640 = 43560 = 4840 = 160 = 4 = 1
72 Arithmetic.
By this measure, land, and husbandmen and gardeners' work
are measured ; also artificers' work, such as board, glass, pave-
ments, plaistermg, wainscoting, tiling, flooring, and every dimen-
sion of length and breadth only.
When three dimensions are concerned, namely, length, breadth,
and depth or thickness, it is called cubic or solid measure, which
is used to measure timber, stone, &c.
The cubic or solid foot, which is 12 inches in length and
breadth and thickness, contains 1728 cubic or solid inches, and
27 solid feet make one solid yard.
Dry^ or Com Measure.
2 Pints make 1 Quart Qt
. 2 Quarts 1 Pottle Pot
2 Pottles 1 Gallon Gal
2 Gallons 1 Peck Pec
4 Pecks 1 Bushel Bu
8 Bushels 1 Quarter Qr
6 Quarters 1 Wey, load, or ton fVey
2 Weys 1 Last L-ast
Pis Gal
8=1 Pec
16= 2= 1 Bu
64 = 8 = 4=1 Qr
512= 64= 32= 8= 1 Wey
2560 = 320 = 160 = 40 = 5 = 1 Last
5120 = 640 = 320 = 80 = 10 = 2 = 1
By this are measured all dry wares, as corn, seeds, roots, fruit,
salt, coals, sand, oysters, &c.
The standard gallon, dry measure, contains 268} cubic or
solid inches, and the com or Winchester bushel 2150| cubic
inches; for the dimensions of the Winchester bushel, by the
statute, are 8 inches deep, and 18^ inches wide or in diameter.
But the coal bushel must be 19| inches in diameter; and 36
bushels, heaped up, make a London chaldron of coals, the
weight of which is 3156 lb Avoirdupois. .
Tahks of Coin, Weight, andMeasuri..
W
Ah an4 Beer Measure*
2 Pinls make
1 Quart Qt
4 Quarts
1 Gallon Gal
36 Gallons
1 Barrel Bar
1 Barrel and a half
1 Hogshead Hhd
2 Barrels
1 Puncheon Pun
2 Hogsheads
1 Butt Butt
2 Butts
1 Tun Tun
Ptfi Qt
2=1 Gfil
8 = 4 = 1 Bfir
288 = 144 = 36 = ]
\ Hhd
432 = 216 = 54 = 1
^= 1 Butt
864 = 432 = 108 = 5
^=2=1
Jiote. The ale gallon contains 282 cubic or solid inches.*
Wine Measure.
2 Pints make
1 Quart Qt
4 Quarts
1 Gallon Gal
42 Gallons
1 Tierce Tier
63 Gallons or 1 J Tierces
1 Hogshead Hhd
2 Tierces
1 Puncheon Pun
2 Hogsheads
1 Pipe or Butt Pi
2 Pipes or 4 Hhds
1 Tun Tun
Pts Qt
2 = 1 Gal
8=4= 1 Tier
336 = 168 = 42 = 1
Hhd
504 = 252 = 63 = li
= 1 Pun
672 = 336 = 84 = 2
= H = 1 Pi
1008 = 504 = 126 = 3
= 2 = li = 1 TVn
2016 = 1008 = 252 = 6
= 4 =3 =2=1
' ^o/e. Bj this are measured all wines, spirits, strong-waters^
cider, mead, perry, vinegar, oil, honey, &c«
The wine gallon contains 231 cubic or solid inches** And it is
' M il. . . ^ » ■ .1 P ■ I I
* The ale i^d wine gallons in England have lately beep reduced
to one of 277 cubic inches.
Arith. 10
74
Arthmttic,
remarkable, tliat the wine and ale gallons have the same prop(XP-
tion to each other, as the troj and avoirdupois pounds have ;
that is, as one pound troy is to one pound avoirdupois^ so is one
wine gallon to one ale gallon.
Of Time.
60 Seconds or 60" make
60 Minutes
24 Hours
7 Days
4 Weeks
13 Months 1 Day 6 Hours, >
or 365 Days 6 Hours )
1
1
1
1
1
Minute
Hour
Day
Week
Month
Jlfor
Hr
Itoy
Wk
Mo
1 Julian Year Yr
Sec
60 =
3600 =
36400 =
604800 =
2419200 =
Mm
1 Hr
60 = 1
1440 = 24 =
10080 = 168 =
Day
1 Wk
7=1 Mo
28 =4 = 1
1 Year
40320 = 672 =
31557600 = 525960 = 8766 = 365^ =
Wk Da Hr Mo Da Hr
Or 52 1 6 = 13 1 6=1 Julian Year
Da Hr M Sec
But 365 5 48 48 = 1 Solar Year.
100. It is evident, that if the several denominations of money,
weight, and measure proceeded in a decimal ratio, the funda-
mental operations might be performed upon these, as upoH
abstract numbers. This may be shown by a few examples in
Federal Money. If it were required to find the sum of {46,85
and j(256,371, we should place the numbers of the same denom-
ination in the same column, and add them together as in whole
numbers; thus,
4685
256371
303221
and the answer may be read off in either or all the denomina-
tions; we may say 30 eagles 3 dollars 22 cents 1 mill, or
Reduction.
75
303 dollars 221 thousandths, or 30323 cents and 1 tenth, or
303221 mills. It is usual to consider the dollars as whole num-
bers, and the following denominations as decimak. The operas-
lion then becomes the same as for decimals*
Add {34,1 23
1,178
78,001
61,789
Sum }1 75,091
Examples.
Add ^456,78
49,83
0,22
7854,394
Sum {8361,224
From {542,76
Subtract 239,481
Rem. 303,279
From 1927,839
Subtract 22,94
Rem. "" 504,899
Multiply {6,347 by {4,532.
Divide {28,764604 by {4,532.
Divide {20 by {2000«
Am. {28,764604.
Ans. {6,347.
Ans. {0,01.
Reduction.
101. When the different denominations do not proceed in a
decimal ratio, they may all be reduced to one denomination, and
then the fundamental operations may be performed upon this, as
upon an abstract number. If, for example, the sum to be oper-
ated upon were £4. 15s. 9d. i\ns may easily be expressed in
pence. As 1 pound is 20 shillings, 4 pounds will be 4 times 20,
or 80 shillings. If to this we ad^ the 15s. we shall have 95s. 9d.
equivalent to the above. But as 1 shilling is equal to 12 pence,
95s. will be equal to 95 times 12, or 1140 pence. Adding 9 to
this, we shall have 1149 pence as an equivalent expression for
£4« 15s. dd. We may now make use of this number as if it had
no relation to money pr any thing else ; and the result obtained
76
Arithmetic*
may be converted again into the different denominations by
versing the process above pnrsued. If it were proposed to mul-
tiply this sum by another nnmber, 37 for instance^ we should
find the product of these two numbers in the usual way } thus,
1149
37
8043
3447
43513
42513 is, therefore, equal to 37 times £4. 15s. 9d« expressed in
pence : to find the number of pounds and shillings contained in
this, we first obtain the number of shillings by dividing it by 12,
which gives 3542, and then the number of pounds by dividing
this last by 20 ; thus,
42513
12
354,2
15
20
65
3542
177
51
14
33
3
9
42513 pence then is equal to 3542 shillings and 9 pence, or to 177
pounds 2 shillings and 9 pence. ViThence 37 times £4. 15s. 9d. is
equal to £l77. 2s. 9d.
It may be remarked, that shillings are converted into pounds h/
separating the right hand figure and dividing those on the left by 2,
prefixing the remainder, if there be one, to the figure separated
for the entire shillings, that remain. This amounts to dividing,
first, by 10 (90), and then that quotient by 2. If 10 shillings
made a pound, dividing by 10 would give the number of pounds,
but as 10 shillings are only half a pound, half this number will
be the number of pounds.
By a method similar to that above given, we reduce other de-
nominations of money and the different denominations of the
several weights and measures to the lowest respectively. If it
were required to find how many grains there are in 2lb. 4oz.
1 7dwt. 5grs. Troy, we should proceed thus,
RtdaefuHk
T7
lb.
oc
dwt*
|W-
2
4
17
4i
12
24
4
28
20
560
17
577
24
2308
1154
13848
6
Ans. 13853
By dividing 13853 by 24, and the quotient thence arising by
20, and this second quotient by 1 2, we shall evidently obtain the
number of pounds, ounces, pennyweights, and grains in 13853
grains* The operation may be seen below*.
13853
24
120
577 20
Ail M . . „
185
168
28
177 24
173
160 — ^
168
4
13
17
Result 2 4 17 5
These examples will be sufficient to establish the foltewing
general rules, namely ^
78 ^ritknutic*
To redtict a compound number to the lowest denomination contain'
ed in t7, multiply the highest by so tnany as one of this denomination
makes of the next lower, and to the product add the number belong'
ing to the next lower ; proceed with each succeeding denomination in a
similar manner, and the last sum will be the number required.
To reduce a number from a lower denomination to a higher,
divide by so many as it takes of this lower denomination to make
one of the higher, and the quotient will be the number of the higher ;
which may be farther reduced in the same manner if there are still
higher denominations, and' the last quotient together with the several
ranainders will be equivalent to the number to be reduced*
Examples for practice*
In 59lb. 13dwt. 5gc. how many grains? Ans. 340157.
In 8012131 grains how many pounds, &c. ?
Ans. 13901b. lloz. 18dwt. 19gr.
In 121/. Os. 9|d. how many halfpence? Ans. 58099.
In 58099 half pence how many pounds &c. ?
Ans. I2ll. Os. 9^d.
In 48 guineas at 28s. each how many 4^ pence ?
Ans. 3584.
In one year of 365d. 5h. 48' 48'' how many seconds ?
Ans. 31556928.
102. When we have occasion to make use of a number con-
sisting of several denominations as an abstract number, instead of
reducing the several parts to the lowest denomination contain-
ed in it, we may reduce all the lower denominations to a frac-
tion of the highest. Taking the sum before used, namely, 4l.
15s. 9d. we reduce the lower denommations to the higher, as
in the last article by division. The number of peace 9, or f , is di-
vided by 12, by multiplying the denominator by this number (54),
we have thus, y'^s. which being added to 15s. or W^s. the whole
number being reduced to the form of a fraction of the same
denominator, we have VV and yVi which being added, make
Vt* This is further reduced to pounds by dividing it by 20,
fbat is. by multiplymg the denominator by 20 (54), which
Meduction^ 79
gives llf. Whence £4. 16s. 9d. is equal to £4^} J, or £ V^V*
This may now be used like any other fraction, and the value of
the result found in the different denominations. If we multiply it
by 37, we shall have £*Ht% or £177^\ ; and £^j\, reduced
to shillings by multiplying the numerator by 20, or dividing the
denominator by this number, gives ^|s. or Sy'yS. or 2s. 9d.
From the above example we may deduce the following general
rules, namely.
To reduce the several parts of a compound number to a fraction
of the highest denomination contained in t/, make the lowest term
the numerator of a fraction^ having for Us denominator the number
which it takes of this denomination to make one of the next higher j
and add to this the next term reduced to a fraction of the same
denomination^ then multiply the denominator of this sum by so many
as make one of the next denonination^ and so on through all the
terms^ and the last sum will be the fraction required,^
y
To find the value of a fraction of a higher denomination in terms of
a lower^ multiply the numerator of the fraction by so many as make
one of the lower denomination^ and divide the product by the denom-
inatorj and the quotient tdHI be the entire number of this denomi-
nation^ the fractional part of which may be still further reduced in
the same manner.
To reduce 2w. Id. 6h. to the fraction of a month.
6h. is j\ of a day, and being added to one day, or f|d. gives
||d. the denominator of which being multiplied by 7, it becomes
tVVw. and being added to 2 weeks or twice Hfw. gives |||w.
If we now multiply the denominator of this by 4, we shall
have fnjf of ^ month, as an equivalent expression for 2w. Id. 6h.
t It will oAen be found more conv^nieDt to reduce the several
parts of the compound number to the lowest denomination, as by the
preceding article for a numerator, and to take for the denominator
90 many of this denomination as it takes to make one of that, t*
which the expression is to be reduced; thus 4/. 15s. 9d. being H49d^
is equal to ^^^L because Id. is ^|^/.
80
AfUhnHiu.
To find the value of 4 of a mile in forlongs, poles, &o.
5
8
40
7
35 "
6
5
40
SOO
7
14
28
60
56
4
H
22
" 7
21
«• «
_ H
1
Ans. 5fur. 28pls. 3| yds.
Reduce 13s. 6d. 2q. to the fraction of a pound.
Ans. £iii, or £H.
Reduce 6fur. 26pls. 3yds. 2ft. to the fraction of a mile.
Ans. H«, or |.
Reduce 7oz. 4pwt. to the fraction of a pound, Troy.
What part 6f a mile is 6fur. 1 6pls. ?
What part of a hogshead is 9 gallons?
What part of a day is tV of a month ?
What part of a penny is tV of a pound ?
What part of a cwt. is f of a pound, ATourdupois ?
What part of a pound is | of a farthing ?
What is the value of | of a pound, Troy ? Ans. 7oz. 4dwt.
What is the value of 4 of a pound. Avoirdupois ?
Ans. 9oz. 24dr.
What is the value of | of a cwt. ? Ans. 3qrs. 3lb. loz. 12|dr.
What is the value of tt of a mile ?
Ans. Ifur. 16pls. 2yds. 1ft. 9fV iu-
Ans. f .
Ans. !•
Ans. 4.
Ans. If.
Ans. V*
Ans. ^i^*
Ans. tiVt*
\
Reiucticn* 81
What is the value of j\ of a day ? Atu. ISh. 55^ ^Sj\'\
The several parts of a compound number may also be re-
duced to the form of a decimal fraction of the highest denomi-
nation contained in it, by first finding the value of the expres-
sion Jn a vulgar fraction, as in the last article, and then reducing
this to a decimal, or more conveniently by changing the terms
to be reduced into decimal parts, and dividing the numerator
instead of multiplying the denominator by the numbers succes-
sively employed in raising them to the required denomination.
If we take the sum already used, namely, £4. 15s. 9d. the
pence, 9, may be written f |, or f JJ, the numerator of which
admits of being divided by 1 2 without a remainder. It is thus
reduced to shillings and becomes yVyS* or O^TSs. which added to
the 15s. makes 15,75s. or reducing the 15 to the same denomi-
nation, VW or VirVoV 5 ^^^ ^18 is reduced to pounds, by
dividing it by 20, the result of which is yVVA? or 0,7875.
4/. 15s. 9d. therefore may be expressed in one denomination,
thus, 4,7875/. and in *this state it may be used like any other
number consisting of an entire and fractional part. If it be
multiplied by 37, we shall have for the product 177,1375/. This
decimal of a pound may be reduced to shillings and pence, by
reversing the above process, or by multiplying successively by
20 and then by 12.'
0,1375
20
2,7500
12
9,0000
The product therefore of 4/. 15s. 9d. by 37 is 177/. Ss. 9d» as
before obtained.
The operation, just explained^ admitis of a more convenient
disposition, as in the following example.
To reduce 19s. 3d. 3q. to the decimal of a pound.
4
12
20
3,00
3,75000
19,312500
0^65625
11
.^
82 Arithmetic*
Proceeding as before, we reduce the farthings, 3^ considered
ais jiiq* to hundredths of a penny by dividing by the figure on the
left, 4, and place the quotient, 75, as a decimal on the right of the
pence ; we then take this sum, considered as rii^* or f H7<1* ^^^^
is, annexing as many ciphers as may be necessary, and divide it by
12, which brings it into decimals of a shilling. Lastly, the shil-
lings and parts of a shilling, 19,3125s. considered as VVVVrsV^*
are reduced to decimals of a pound by dividing by 20, which
gives the result above found.
We may proceed in a similar manner with other denomina-
tions of money and with those of the several weights and meas-
ures. One example in these will suffice as an illustration of the
method.
To reduce I7pls. 1ft. Sin. to the decimal of a mUe.
12
16,5
320
6
1,5
17,09
-f 1
0,0053if53i|&€.
The decimal in this, as in many otner cases, becomes period-*
ical. (97).
From what has been said, the following rules are sufficiendy
evident. To reduce a number from a lomer denomination to the
decinilil of a higher j we first change t/, or suppose it to be changed
into a fraction^ having 10, or some multiple of lO^for its denominor
tor^ and divide the numerator by so many as make one of this
higher denomination^ and the quotient is the required decimal ; vohich^
together with the whole number of this denomination^ may again be
converted into a fraction^ having 10 or a multiple of 10 for its de*
nominator, and thus by division be reduced to a still higher name,
mid so on,
AlsiOy to reduce a decimal of a higher denomination to a lower,
we multiply it by so many as one makes of this lower, and those
figures which remain %n the left of the comma, when the proper
number is separated for decimals (91), will constitute the whole
number of this denomi7iation, the decimal part of which may fc« still
further reduced, if there be lower denominations, by multiplying it
Jby the number which one makes of the next den&fmnaiion, and so on*
Reduetim. 99
It may be proper to add in this place, that shillings, pence, and
farthings may readily be converted into the fraction of a pound,
and the fraction of a pound reduced to shillings, pence, and far-
things, without having recourse to the above rules. As shillings
are so many twentieths of a pound, by dividing any giving num-
ber of shillings by 2, we convert them into decimals of a pound,
thus, 15s. which may be written i|/. or ^f |/. being divided
by 2, give 75 hundredths, or 0,75 of a pound. Also, as farthings
are so many 960ths of a pound, one pound being equal to 960
farthings, the pence converted into farthings and united with
those of this denomination, may be written as so many 960ths of
a pound. If now we increase the numerator and denominator
one twenty fourth part, we shall convert the denominator into
thousandths, and the numerator will become a Yecimal.
Whence, to convert shillings^ pence^ and farthings^ into the decimal
of a pounds divide the shillings by 2, adding a • cipher whefi neces-
jary, and let the quotient occupy the Jirsi place^ or first and second^
if there be two figures j and let the farthings^ contained in the pence
and farthings, be considered as so many thousandths, increasing the
number by one, when the number is nearer 24 than 0, and by 2, when
it is nearer 48 than 24, and so on*
Thus, to reduce 15s. 9d. to the decimal of a ppuQd, we have,
0,75
37
0,787
This result, it will be remarked, is not exactly the same as that
obtained by the other method ; the reason is, that we have increas-
ed the number of farthings, 36^ by only one^ whereas allowing
ane for every 24, we ought to have increased it one and a half.
Adding, therefore, a half, or 5 units of the next lower order, we
shall have 0,7875, as before.
On the other hand, the decimal of a pound is converted into the •
lower denominations, or its value is found in shillings, pence, and
farthings, by doubling the first figure for shillings, increasing it by
one, when the second figure is 6, or more than 5, and considering
what remains in the second and third places, as farthings, after
having diminished them entfor every 24.
• •
84 Arithmetic^
*
In addition to the rales that have been given, it may be observe
ed, that in those cases, where it is required to reduce a number
from one denomination to another, when the two denominations
are not commensurable or when one will not exactly divide the
other, it will be found most convenient, as a general rule, to re-
duce the one, or both, when it is necessary, to parts so small, that
a certain number of the one will exactly make a unit of the
other. If it were required, for instance, to reduce pounds to
dollars, as a pound does not contain an exact number of dollars
without a fraction, we first convert the pounds into shillings, and
then, as a certain number of shillings make a dollar, by divid-
ing the shillings by this number, we shall find the number of
dollars required. A similar method may be pursued in other
cases of a like nfture, as may be seen in the following examples.
In V7B guineas at 28s. each, how many crowns at 6s. 8d. ?
6s. 8d.
2
178
28
5980,8
48
80
747
Od.
1424
356 .
4984
12
59808
Ans. 747 crowns and 4 shillingst.
In this case, I reduce both the guineas and the crown .to pence,
and then divide the former result by the latter. In dividing by 80,
I first separate one figure on the right of the dividend for a deci-
mal, which is the same as dividing it by 10, and then divide the
figures on the left, or the quotient, by 8 (47), joining what re-
mains as tens to the figures separated, to form the entire remain-
der, which is reduced back to* the original denomination.
t Questions of this kind may of)en .he conveniently performed hy
fractions; thus, 178 guineas, or 4984s. divided by Gs, 8d. or 6fs, or,
reducing the wliole number to the form of a fraction, '^^. becomes
*V* multiplied by ^ (74), or ^*jV^ "^^ ^*|**', which is equal to
747^ ; and, \\^ or j, of Bs. 8d. is 3 times \ of 80d. or 48d. or 4fl.
RtducHo9i»
85
To reduce 137 five franc pieces to pounds, shillings, &c. the
franc being valued at ^,1866.
0,1866
5
0,9330
137
6531
«799
933
127,821
6
766,926
76,6926
20
38,3463
20
6,926
12
11,112
4
448
Ans» 38/. 6s. lid. O^q. nearly.
Examples for practice.
Reduce 7s. 9|d. fo the decimal of a pound. Ans. 0,390625.
Reduce 3qrs. 2nji. to the decimal of a yard. Ans, 0,875.
Find the value oi 0,85251/. in shillings, pence, &c.
Ans. 17s. Od. 2|q. nearly.
Reduce 241/. 18s. 9d. to federal money. Ans. J806,4583 &c.
Find the value of 0,42857 of a month.
Ans. Iw. 4d. 23h. 59' 56".
Required the cin umfercnce of the earth in English statute
miles, a degree being estimated at 57008 toisest*
Ans. 24855,488.
We have given rules for reducing a compound number from
one denomination to another, as we shall have frequent occasion
in what follows for making these reductions. They are not,
however, necessary, except in particular cases, previously to per-
forming the fundamental operations. The several denoinina-
tions of a compound number may be regarded like the different
orders of units in a simple one, that is, the number or numbers of
t A toise or French U thorn is equal to 6 French feet, and a French
foot is equal to 12,7893 English inches.
f4''
86 Arithmetic.
each denomination may be made the subject of a distinct opera-
tion, the result of which, being reduced when necessary, may be
united to the next, and so on through all the denominations*
Addition of Compound Numbers.
103. The addition of compound numbers depends on the same
principles as that of simple numbers, the object being simply
to unite parts of the same denomination, and when a num-
ber of these are found, sufficient to form one, or more than one
of a higher, these last are retained to be united to others of the
same denomination in the given numbers ; as in simple addition
the tens are carried from one column to the next column on the
left. JVe rnust^ thtn^ place the compound numbers^ that art to be
added^ in such a manner^ that their units^ or parts of the sarnt
name^ may stand under each other ; we must then find separately
the sum of each Column^ always recollecting how many parts of
§ach denomination it takes to make one of the next high$r» See the
following example in pounds, shillings, and penee.
£
s.
d.
984
12
8
38
6
9
1413
14
10
319
18
2
2756 12 5
First, adding together the pence, because they are the parts of
the least value, and taking together both the units and tens of
this denomination, we find 29; but as 12 pence make a shil-
ling, this sum amounts to 2 shillings and 5 pence; we then
write down only the 5 pence, and retain the shillings in order to
unite them to the column to which they belong.
Next, we add separately the units and the tens of ^e next de-
nomination ; the. first give, by joining to them the 2 shillings re-
served from the pence, 22 ; we write down only the t^o units and
retain the two tens for the next coluoui, the son of which, by this
Addition ofCon^^ound J^umbers* 87
meaixs, amounts to 5 tens, but as the pound, made up of 20 shil-
lings, contains 2 tens, we obtain the number of pounds result-
ing firom the shillings, by dividing the tens of these last by 2 ;
the quotient is 2, and the remainder 1, which last is written
under the column to which it belongs, while the pounds are re-
served for the next column on the left; as this column is the last
the operation is performed as in simple numbers, and the whole
sum is found to be 2756/. 12s. 5d«
The method of proving the addition of compound numbers is
derived from the same principles, as that for simple numbers,
and is performed in the same manner, care being taken in passing
from one denomination to another, to substitute instead of the
decimal ratio, the value of each part in the terms of that, which
•follows it on the right. Let there be, for example,
£
si
d.
d84
12
8
38
6
9
1413
14
10
319
18
2
2756
12
5
1122 22
The operation on the pounds is performed according to the
rule of ailicle 19 ; then we change the two pounds into tens of
shillings, and obtain 4 of these tens, which, joined to that written
under the column, makes 5, from which we subtract the 3 units
of this column, and place the remainder, 2, underneath, counting
it as tens with regard to the next column. There still remain
2 shillings, which must be reduced to pence ; adding the result,
24 pence, to the 5 that are written, we have a total of 29, which
must be again obtained by the addition of all the pence, as these
are the parts of the lowest denomination in the question. This
really happens, and proves the operation to be rights
88 ArithmetH*
Examples*
£
8.
d.
£
s.
d.
17
13
4
84
17
H
13
10
2
7^
1-3
H
10
17
3 .
51
17
8|
8
8
7
20
10
lOy
3
3
4
17
15
4i
8
8
10
10
11
£
s.
d.
175
10
10
101
13
Hi
89
18
10
75
12
2i
3
3
3|
1
I
Sum 54 1 4 261 5 8^ 452 19 2|
-fecrt-^'^-O -« -S 30 232 n3
lb.
OS. dwt.
gr.
lb.
OS. dwt.
gr-
lb. OS. dwt
P*-
17
3 15
11
14
10 13
20
27 10 17
18
13
2 13
13
13
10 18
21
17 10 13
13
15
3 14
14
14
10 10
10
13 11 13
1
13
10
10
1 2
3
10 1
2
12
1
17 ■
1
4 4
4
4 4 3
3
13
14
1 19
2
1
cwt.
qr. lb. OS.
dr.
T. cwt,
. qr. lb. OS.
dr.
T. cwt. qr. lb. os.
dr.
15
2 15 15
15
2 17
3 13 8
7
3 13 2 10 7
7
13
2 17 13
14
2 13
3 14 8
8
2 14 1 17 6
6
12
2 13 14
14
1 16
10
5
4 17 14
6
10.
1 17 15
2 13
1
7
2 13 12 7
7
12
1 10
10
1 14
1 1 2
2
3 13 10 4
4
10
1 12 1
7
4 16
1 7 7
5
5 2 12 8
8
Mis.
fur. pol. yd 1
't. in.
Mis. fur. pol. yd. ft.
in.
MU. for pol. yd. ft.
iu.
37
3 14 2 1
[ 5
23
2 13 1 1
4
28 3 7 2
7
28
4 17 3 2 10
39
1 17 2 2
10
30 1
7
17
4 4 3 1
I 2
28
1 14 2 2
27 6 30 2 2
10
5 6 3:
I 7
48
1 17 2 2
7
7 6 20 2 1
29
2 2 2
3
37
1 29
3
5 2 2
10
30
4
2
2
20 2
1
7 10 2
2
Subtraction of CompoumI Numbers, 89
Subtraction of Compound Numbers.
104. This operation is performed in the same way as the sub-
traction of simple numbers, except with regard to the number
which it is necessary to borrow from the higher denominations,
in order to perform the partial subtractions, when the lower
number exceeds the upper. For instance,
£
s.
d.
from
795
3
take
684
17
4
Difference
110
5
8
In performing this example, it is necessary to borrow, from the
column of shillings, 1 shilling or 1 2 pence, in order to effect the
subtraction of the lower number, 4, and we have for a remainder
8 pence. There now remain in the upper number of the column of
shillings only 2 ; it is necessary therefore to borrow, from that of
pounds, 1 pound or 20 shillings, we thus make it 22, of which,
when the lower number, 1 7, is subtracted, 5 remain ; we must
now proceed to the column of pounds, remembering to count the
upper number less by unity, and finish the operation as in the
case of simple numbers.
The method of proving subtraction of compound numbers, like
that for simple numbers, consists in adding the difference to the
less of the two numbers.
Examples for practice*
£
275
176
s.
13
16
d.
4
6
£
454
276
177
s.
14
17
16
d.
2|
£
274
85
188
274
8.
14
15
18
14
Rem. 98
16
10
6*
Proof 275
13
4
12
454
14
2J
2f
90 Ariikmttk.
lb. oz. dwt« gr. lb. oz. dwt. gr. lb. oz. dwt. gr.
7 3 14 11 27 2 10 20 29 3 14 5
3 7 15 20 20 3 5 21 20 7 15 7
Rem.
Proof
cwt. qr. lb. oz, dr. cwt. qr. lb. o«. dr. cwt. qr. lb. oz. dr.
5 17 5 9 22 2 13 4 8 21 1 7 6 13
8 3 21 1 7 20 1 17 6 6 13 8 8 14
Rem.
Proof
Mis. far. pel. yd. ft. in. Mil. fur. poL yd. ft. in. Mis. fur. pol. yd. ft. io.
14 3 17 1 2 1 70 7 13 1 1 2 70 3 10 7
10 7 80 2 10 20 14 2 2 7 17 3 11 1 1 3
Rem.
Proof
m. w. d. h. '
17 2 5 17 26
10 18 18
m. w. d. h. '
37 1 13 1
15 2 15 14
m. w. d. h.
71 5
17 6 5 7
Rem.
Proof
•
Multiplication of Compound Numbers. '
105. We have seen, that a number consisting of several denom*
(nations may be reduced to a single one, either the lowest or the
highest of those contained in it, in which state it admits of being
used as an abstract number. But when it is required to find the
product of two numbers, one of which only is compound, the sim*
Multiplication of Compound Jf umbers. 91
plest method is, to consider the multiplication of each denomina-
tion of the compound number by the simple factor, as a distinct
question, and the several results, thus obtained, will be the total
product sought. If it were pro[)osed, for example, to multiply
?/• I4si 7d. 3q. by 9, it may be done thus.
£
s.
d.
q
7
14
7
3
9
9
9
9
63 126 63 27
and 63/. 126s. 63d. 27q. is evidently 9 times the proposed sum,
because it is 9 times each of the parts, which compose this sum.
But 27q. is equal to 6d. 3q. and adding the 6d. to the 63d. we
have 69d. equal to 5s. 9d. adding the 5s. to the 126s. we obtain
131s. equal to 6/. J Is. and lastly, adding the 6/. to the 63/. we
have 69/. Us. 9d. 3q. equal to the above result, and equal to the
product of
7/. 14s. 7d. 3q. by 9.
Instead of finding the several products first, and then reducing
them, we may make the reductions after each multiplication,
putting down what remains of this denomination, and carrying
fonvard the quotient, thus obtained, to be united to the next
higher product.
Hence, to multiply two numbers together^ one of which is com-
pound^ make the compound number the multiplicand and the simple
number the multiplier^ and beginning with the lowest denomination
of the multipUcand^ multiply it by the multiplier^ and divide the pro-
duct by the number which it takes to make one of the next superior
denomination ; putting down the remainder^ add the quotient to the
product of the next denomination by the multiplier, reduce this sum^
jmtting down the remainder and reserving the quotient, as before,
and proceed in this manner through all the denominations to the
last, which is to be multiplied like a simple number.
When the multiplier exceeds 12, that is, when it is so large
that it is inconvenient to multiply by the whole at once, the
shortest method is to resolve it, if it can be done, into two or
9$ Arithmetic.
more factors, and to multiply first by one and then that product
by the other, aad so on, as in the following example. Let the
two numbers be £4 13s. 3d. and 18.
£
s.
d.
4
13
3
9
ft
41
19
3
2
83 18 6
Here we first find 9 times the multiplicand, or £41 19s. 3d.
and then take twice this product, which will evidcBtly be twice
9 or 18 times the original multiplicand (82). Instead of multi-
plying by 9, we might multiply first by 3 and then that product
by 3, which would give the same result; also the multiplier 18
might be resolved into 3 and 6, which would give the same pr(^
duct as the above. If we multiply £83 18s. 6d. by 7,
£
s.
d.
83
18
6
7
587 9 6
we shall have the product of the original multiplicand by 7 times
18 or 126.
If the multiplier were 105, it might be resolved into 7, 3, and
5, and the product be found as above.
But it frequently happens, that the multiplier cannot be re-
solved in this way into factors. When this is the case, we may
take the number nearest to it, which can be so resolved, and
find the product of the multiplicand by this number, as already
described, and then add or subtract so many times the multipli-
cand, as this number falls short, or exceeds the given multiplier,
an -J the result will be the product sought. Let there be i&l 7s.
8d. to be multiplied by 1 7.
Multiplication of Compound J^umbers, 93
£
8.
d.
1
7
8
4
5
10
8
4
22
2
8
1
7
8
Product £23 10 4
In the first place, I find the product of £l 7s. 8d. by 16, which
is £22 2s. 8d. and to this 1 add once the multiplicand and this
sum £23 10s. 4d. is evidently equal to 17 times the multiplicand.
106. It may be observed, that in those cases, where the decrease
of value from one denomination to another is according to
the same law throughout, that is, where it takes the same number
of a lower denomination to make one of the next higher through
all the denominations, the multiplication of one compound number
by another may be performed in a manner similar to what takes
place with regard to abstract numbers.
This regular gradation is sometimes preserved in the denom-
inations that succeed to feet, in long measure, 1 inch or prime
being considered as equal to l^seconds^ and 1 second to 12 thirds^
and so on, the several denominations after feet being distinguish-
ed by one, two, &c. accents, thus,
lOf. 4' 6" 10'".
If it were required to find the product of 2f. 4' by 3f. lO', we
should proceed as below.
2f. 4'
3 10
1 11
7
8 ,11 4"
The 4 inches or primes may be considered with reference to
the denomination of feet, as 4 twelfths, or 7^9 ^^^ ^^^ 10 inches
B4 Arithmetic.
as II, the product of which is yY^, or || of y^, or 40", which
reduced gives 3' 4''; putting down the 4", we reserve the 3' to be
added to the product of 2 feet by 10', or ||, which product is f |
of a foot, to which 3 being added, we have f-|f. or If. and 11' ;
next muhiplying 4' or y*^ by 3, we have || or 1, which added to
the product of 2 by 3 gives 7. Taking the sum of these results,
we have 8f, 11' 4", for the product of 2f. 4" by 3f. 10'. The
method here pursued may be extended to those cases, where
there is a greater number of denominations.
Whence, to multiply one number consisting of fett^ primes^
seconds, <$rc. by another of the same kind, havir^ placed the several
terms of the multiplier under the corresponding ones of the multi^
plicand, multiply the whole multiplicand by the several terms of
the multiplier successively according to the rule of the last article^
placing the first term of each of the partial products under its res*
pective multiplier, and find the sum of the several columns, observing
to carry one for every twelve in each part of the operation ; then the
first number on the left will be feet, and the second primes, and the
third seconds, and so on regularly to the lastA
Examples for practice.
Multiply £1 lis. 6d. 2q. by 5. Ans. £7 ITs. 8d. 2q.
Multiply 7s. 4A 3q. by 24. Ans. £B 17s. 6d.
Multiply £1 17s. 6d. by 63. Ans. £118 2s. 6d.
Multiply 17s. 9d. by 47. Ans. £41 14s. 3d.
Multiply £1 28. 3d. by 117. Ans. £130 3s. 3d.
What is the value of 119 yards of cloth at £2 4s. 3d. per
yard? Ans. £263 5s. 9d.
t The above article relates to what is commonly called duodeci-
fnals» The operation is ordinarily performed bj beginning with the
highest denomination of the multiplier^ and disposing of the several
products as in the first example below. The result is evidently the
same, whichever method is pursued, as may be seen by comparing
Divxston of Compmnd Kuwhtrs. 95
What is the value of 9cwt. of cheese at £1 lis. 5d. per cwt ?
Ans. £14 2s. 9d.
What is the value of 96 quarters of rye at £1 3s. 4d. per
quarter? Ans. £112,
What is the weight of 7hhds. of sugar, each weighing 9cwt.
3qrs. 12lb? Ans. 69cwt,
In the Lunar circle of 19 years, of 365d. 5h. 48' 48" each,
how many days, &c. ? Ans. 6939d. 14h. 27' 12",
Multiply 14f. 9* by 4f, 6'. Ans. 66f. 4' 6",
Multiply 4f. r 8" by 9f, 6'. Ana. 44f. 0' 10".
Required the content of a floor 48f. 6' long and 24f. 2^ broad.
Am. Il76f. r 6".
What is the number of square feet &c. in a marble slab,
whose length is 5f. 7' and breadth If, 10'? Ans. lOf. 2' 10'.
Division of Compound Numbers.
107. A COMPOUND number may be divided by a simple num-
ber, by regarding each of the terms of the former, as forming a
distinct dividend. If we take the product found in article 105,
namely, £63 126s. 63d. 27q. and divide it by the multiplier 9,
this example with that of the same question on the right, performed
according to the rule in the text This last arrangement seems to
be preferable, as it is more strictly conformable to what takes place
in the multiplication of numbers accompanied by decimals.
f. ' " f. ' "
10 4 5 10 4 5
7 8 1 7 ' 8 1
72 6 11 5 2 2 6
6 10 11 4"' 6 10 11 4
6 2 2 &'" 72 6 11
79 II 6 6 79f ir 0" 6"'
iHI0
96 Aritkmelicm
we shall evidently comeback to the multiplicand, iC7 14s. 7d. 3q.
We arrive at the same result also, by dividing the above sum re-
duced, or £69 lis. 9d. 3q. for we obtain one 9th of each of the
several parts that compose the number, the sum of which must be
one 9th of the whole. But since, in this case, each term of the
dividend is not exactly divisible by the divisor, instead of employ-
ing a fraction, wc reduce what remains, and add it to the next
lower denomination, and then divide the sum thus formed, by the
divisor. The operation may be seen below.
£69
lis.
9d.
3q.
9
63
6
20
£7 148, 7d. 3q.
131
41
36
5
12
69
63
6
4
27
27
Whence, to divide a number consisting of different denominations
hf a simple number^ divide the highest term of the compound num-
her by the divisor, reduce the remainder to the next lotoer denomr
nation, adding to it the number of this denomination, and divide the
9um by the divisor, reducing the remainder, as before, and proceed in
Division of Compound Jf umbers.
97
this way Oirough all the denominations to the last^ the remainder
of whichf if there be one, must have its quotient represented in the
form of a fraction by placing the divisor under it. The sum of the
several quotients, thus obtained^ voill be the whole quotient required.
When the divisor is large and can be resolved into two or
more simple factors, we may divide first by one of these factors,
and then that quotient by another, and so on, and the last quo-
tient will be the same as that which would have been obtained
by using a whole divisor in a single operation. Taking the
result of the example in the corresponding case of multiplication,
we proceed thus,
£83
18s. 6
d.
2
8
£41
19s.
3d.
9
3
2
36
£4
13s.
3d.
<—
5
1
20
20
—
119
38
9
2
....
—
29
18
27
18
—
—
2
12
6
....
6
27
—
27
By dividing £83 18s. 6d. by 2, we obtain one half of this sum,
which, being divided by 9, must give one 9th of one half, or one
18th of the whole. The first operation may be considered as
separating the dividend into two equal parts, and the second as
distributing each of these into nine equal parts, the number of
parts therefore will be 18, and being equal,, one of them must be
one 18th of the whole.
But when the divisor cannot be thus resolved, the operation
must be performed by dividing by the whole at once. If the
13
'<»**■
98 ArUhmetic*
quotient, which we are seeking, were known, by adding it to^ or
subtracting frdm it, the dividend a certain number of times
increasing or diminishing the divisor at the same time by as
many units, we might change the question into one, 'whose divi-
sor would admit of being resolved into factors, which would
give 'the same quotient; we should thus preserve the anology
which exists between the multiplication and division of compound
numbers. But this cannot be done, as it supposes that to be
]<nown, which is the object of the operation.
Multiplication and division, where compound numbers are
concerned, mutually prove each other, as in the case of simple
numbers. This may be seen by comparing the examples, which
are given at length to illustrate these rules.
Examples for practice.
Divide £821 17s. 9fd. by 4. Ans. £205 9s. 5Jd.
Divide £28 2s. l|d. by 6. Ans. £4 13s. 8Jd.
Divide £57 3s. 7d. by 35. Ans. £\ 12s. 8d.
Divide £23 15s. T^d. by 37. Ans. 12s. lOJd.
Divide 1061cwt. 2qrs. by 28. Ans. 37cwt.3qrs. 18lb.
Divide 375mls. 2fur. 7pls. 2yds. 1ft. 2m. by 39.
Ans. 9mls. 4fur. 39pls. 3ft. 8in«
if 9 yards of cloth cost £4 3s. 7|d. what is it per yard 7
Ans. 9s. 3d. 2q.
If a hogshead of wine cost £33 1 2s. what is it per gallon ?
Ans. lOs. 8d.
If a dozen silver spoons weigh 3lb. 2oz. 13pwt. 12grs. whstt
is the weight of each spoon? Ans. 3oz« 4pwt. llgrs.
If a person^s income be £150 a year, what is it per day ?
Ans. 8s. 2|d. nearly.
A capital of £223 16s. 8d. being divided into 96 shares,
what is the value of a share? ' Ans. £2 6s. 7|d.
Proportion.
108. We have shown in the preceding part of this work, the dif-
ferent methods' necessary for performing on all numbers, whethec
Proportion. 99
whole or fractional, or consisting of different denominations,
the four fundamental operations of arithmetic, namely, addition,
rabtraction, multiplication, and division ; and all questions rela-
tive to numbers ought to be regarded as solved, when, by an
attentive examination of the manner in which they arc stated,
they can be reduced to some one of these operations. (%nsc«
quently, we might here terminate all that is to be said on arith-
metic, for what remains belongs, properly speaking, to the prov-
ince of algebra. We shall, nevertheless, for the sake of exer-
cising the learner, now resolve some questions which will prepare
him for algebraic analysis, and make him acquainted with
a very important theory, that of ratios and proportions, whitji
is ordinarily comprehended in arithmetic.
109. A piece of chth 13 yards long was sold for 130 dollars ;
what will be the price of a piece of the same cloth 1 8 yards long ?
It is plain, that if we knew the price of one yard of the cloth
that was sold, we might repeat this price 18 times, and the
result would be the price of the piece 18 yards long. Now
since 13 yards cost 130 dollars, one yard must have cost the
thirteenth part of 130 dollars, or VV > performing the division,
we find for the result 10 dollars, and multiplying this number by
18 we have 180 dollars for the answer; which is the true cost of
the piece 18 yards long.
A courier^ who travels always at the same rate^ having gone 5
leagues in 3 hours^ how many will he go in 1 1 hours f
Reasoning as in the last example, we see, that the courier
goes in one hour | of 5 leagues, or f , and consequently, in 1 1
hours he will go 1 1 times as much, or f of a league multiplied
by 11, or V, that is 18 leagues and 1 mile.
In how many hours loill the courier of the precedeng question go
32 leagues ?
We see,'that if we knew the time he would occupy In going one
leagye, we should have only to repeat this number 23 times, and
the result would be the number of hours required. Now the
courier, requiring 3 hours to go 5 leagues, will require only
\ of the time, | of an hour,* to go one league ; this number,
multiplied by 22, dves V or 13 hours and }, that is, 13 hours
and 13 minutes.
100 Arithmetic^
1 1 0. Wc have discovered the unknown quantities by an analy-
sis of each of the preceding statements, but the known numbers
and those required depend upon each other in a manner, that it
would be well to examine.
To do this, let us resume the first question, in which it was re-
quired to find the price of 18 yards of cloth, of which 13 cost
130 dollars.
It is plain, that the price of this piece would be double, if the
number of yards it contained were double that of the first ; that
if the number of yards were triple, the price would be triple also,
and so on ; also that for the half or two thirds of the piece we
should have to pay only one half or two thirds of the whole price*
According to what is here said, which all those, who understand
the meaning of the terms, will readily admit, we see that if there
be two pieces of the same cloth, the price of the second ought to
contain that of the first, as many times as the length of the
second contains the length of the first ; and this circumstance is
stated in saying, that the prices are in proportion to the lengths
or have the same relation to each other as the lengths.
This example will serve to establish the meaning of several
terms which frequently occur.
•111. The relation of the lengths is the number, whether whole
or fractional, which denotes how many times one of the lengths
contains the other. If the first piece had 4 yards and the second
8, the relation, or ratio, of the former to the latter would be 9,
because 8 contains 4 twice. In the above example, the first piece
had 13 yards and the second 18 ; the ratio of the former to the
latter is then ||, or 1 y\. In genaral, the relation or ratio of two
numbers is the quotient arising from dividing one by the other •
As the prices have the same relation to each other, that the
lengths have, 180 divided hf 130 must give |f for a quotient,
which is the case ; for in reducing ||{ to its most simple terms,
wc get j|.
The four nuoibers, 13, IS, 130, 180, written in this order, arc
then such that the second contains the first as many times as the
fourth contains the third, and tbus*they form what is called a
proportion.
We see also, that a proportion is the combination of ttoo equal
ratios^
Prepwriion* 101
We may observe, in this connexion, that a relation is not
changed by multiplying each of its terms by the same number ;
and this is plain, because a relation, being nothing but the quo-
tient of a division, may always be expressed in a fractional form.
Thus the relation j-f is the same as |f f .
The same considerations apply also to the second example.
The courier, who went 5 leagues in 3 hours, would go twice as
far in double that time, three times as far in triple that time ;
thus 1 1 hours, the thne spent by the courier in going 1 8 leagues
and I, or y of a league, ought to contain 3 hours, the time re-
quired in going 5 leagues, as often as Y contains 5.
The four numbers 5, Y? ^9 11? ^^^ ^^^^ ^^ proportion ; and in
reality if we divide Y by 5, we get f |, a result equivalent to y .
It will now be easy to recognise all the cases, where there may
be a proportion between the four numbers.
112. To denote that there is a proportion between the num-
bers 13, 18, 130, and 180, they are written thus,
13: 18 :: 130: 180,
which is read IS is to 18 a^ 130 t^ to 180 ; that is, 13 is the same
part of 18 that 130 is of 180, or that 13 is contained in 18 as
many times as 130 is in 180, or lastly that the relation of 18 to
1 3 is the same as that of 1 80 to 1 30.
The first term of a relation is called the antecedent, and the
second the consequent. In a proporion there are two antecedents
and two consequents^ viz. the antecedent of the first relation and
that of the second ; the consequent of the first relation and that
of the second. In the proportion 13 : 18 :: 130 : 180, the ante-
cedents arc 13, 130 ; the consequents 18 and 180.
We shall in future take the consequent for the numerator, and
the antecedent for the denominator of the fraction which ex-
presses the relation.
113. To ascertain that there is a proportion between the four
numbers 13, 18, 130, and 180, we roust see if the fractions |f
and f 1^ be equal, and to do this, we reduce the second to its most
simple terms; but this verification may also be made by con-
sidering, that if, as is supposed bj^ the nature of proportion, the
two fractions |f and f |f be equal, it follows that, by reducing
them to the same dcnoDiinator, the numerator of the one will be-
102 Ariihmeiicm
come equal to that of the other, and that, consequently, 18 multi-
plied by 130 will give the same product as 180 by 13. This is
actually the case, and the reasoning by which it is showR, being
independent of the particular values of the numbers, proves,
that, if four numbers be in proportion^ the product of the first and
lasty or of the two extremes, is eqtial to the product of the second and
thirdf or of the two tneansm
We see at the same time, that, if the four given numbers were
not in proportion, they would not have the abovementioned pro*
perty ; for the fraction, which expresses the first ratio, not being
equivalent to that which expresses the second, the numerator of
the one will not be equal to that of the other, when they are re-
duced to a common denominator.
114. The first consequence, naturally drawn from what has
been said, is, that the order of the terms of a proportion may be
changed, provided they be so placed, that the product of the ex-
tremes shall be equal to that of the means. In the proportion
13 : 18 : : 130 : 180, the following arrangements may be made^
13: 18:: 130: 180
13 : 130: : 18 : 180
180:130:: 18: 13
180: 18:: 130: 13
18: 13: : 180: 130
18:180:: 13:130
130 : 13 : : 180 : 18
130 : 180 : : 13 : 18
for in each one of these, the product of the extremes is formed of
the same factors, and the product of the means of the same fac-
tors. The second arrangement, in which the means have chang-
ed places with each other, is one of those that most frequently
occur.*
* It may be observed, that the proportion 13:130::18:180
might have been at once presented under this form, according to the
solution of the question in article 109 ; for the value of a yard of
cloth may be ascertained in two ways, namely, by dividing the price
of the piece of 13 yards by 13, or by dividing the price of 18 yards
by 18 ; it follows then tha^ the price of the first mast contain 13 as
Proportum* 103
115. This change shows that we may either multiply or divide
the two antecedents, or the two consequents, by the same num-
ber, without destroying the proportion. For this change makes
the two antecedents to constitute the first relation, and the two
consequents, the second. If, for instance, 55 : 21 : : 165 : 63,
changing the places of the means we should have,
55: 165:: 31 : 63;
we might now divide the terms, which form the first relation, by
5, (111) which would give 11 : 33 : : 31 : 63, changing again the
places of the means, we should have 11 : 31 : : 33 : 63, a propor-
tion which is true in itself, and which does not differ from the
given proportion, except in having had its two antecedents
divided by 5.
116. Since the product of the extremes is equal to that of
the means, one product may be taken for the other, and, as in di-
viding the product of the extremes, by one extreme, we must ne-
cessarily find the other as the quotient, consequently^ in dividing
by <me extreme the product of the means^ we shall find the other
extreme. For the same reason, if we divide Jhe product of the ex-
tremes by one of the means, we shall find the other mean*
We can then find any one term of a proportion, when we know
the other three, for the term sought must be either one of the
extremes or one of the means.
The question of article (109) may be resolved by one of these
rules. Thus, when we have perceived that the prices of the
two pieces are in the proportion of the number of yards contain-
ed in each, we write the proportion in this manner,
13 : 18 : : 130 : a;.
many times as the price of the second contains 18 ; we shall then
have 13 : 130 : : 18 : 180. We may reason in the same manner
with respect to the 2nd question In the article above referred to, as
well as with respect to ail others of the like kind, and thence derive
proportions ; but the method adopted in article 109 seemed preferable,
because it leads us to compare together numbers of the same denom-
ination, whilst by the others we compare prices, which are sums of
money, with yards, which are measures of length ; and this cannot
be done without reducing them both to abstract numbers.
104 Anthmehc.
putting the letter x instead of the required price of 18 yards ;
and we find the price, which is one of the extremes, by multiply-
ing together the two means, 18 and 130, which makes 3340, and
dividing this product by the known extreme, 13, we obtain, for
the result, 180.
The operation, by which, when any three terms of a propor-
tion are given, we find the fourth, is called the Ruh of Thru,
Writers on arithmetic have distinguished it into several kinds,
but this is unnecessary, when the nature of proportion and
the enunciation of the question are well understood ; as a few
examples will sufficiently show.
117. A person having travelled 317,5 miles in 9 days; it is
asked, how long he will be in travelling 423,9 miles, he being
supposed to travel at the same rate-
In this question the unknown quantity is the number of days,
which ought to contain the 9 days spent in going 217,5 miles,
as many times as 423,9 contains 217,5 ; we thus get the following
proportion ;
dayi
217,5 : 423,9 : : 9 : x, and we find for x, 17,54 nearly.
118. AH the difficulty in these questions consists in the man-
ner of stating the proportion. The following rules will be suffi-
cient to guide the learner in all cases. .
Among the four numbers which constitute a proportion, there
are two of the same kind, and two others also of the same kind,
but different from the first two. In the preceding example, two
of the terms are miles, and the other two, days.
First, then, it is necessary to distinguish the two terms of
each kind, and when this is done, we shall necessarily have the
quotient of the greatest term of the second kind by the smallest
of the same kind, equal to the quotient of the greatest term of the
first kind by the smallest of the same kind, which will give us
this proportion ;
Proportion*^ 105 »
the smaller term of the first kind
15.
to the larger of the Same kind
as
the smaller term of the second kind ^
w
to the larger of this kind.
In the preceding example this rule immediately givea
217,5 : 423,9 : : 9 : a?,
for the unknown term ought to be greater than 9, since a greatei*
number of days will be necessary to complete a longer journey,
119. If it were required to find how many days it would take
27 men to perform a pifcc of work, which 15 men, working at
the same rate, would do in 1 8 days ; we see that the days should
be less in proportion as the number of men is greater, and recip-
rocally. There is still a proportion in this case, but the order of
the terms is inverted ; for, if the number of workmen in the
second set were triple of that in the first, they woul4 require
only one third of the time. The first Ifumber •£ days then
would contain the second as many times, Hi lk€fee«nd number
of workmen would contain the first. "Dnf^Milr of the terms
being the reverse of that $ssigned to thenrlly the enunciation of
the question, we say, thaft the number of workmen is in the
inoerse ratio of the number of days. If we compare the two first
and the two last, in the order in which they present themselves,
the ratio of the former will be 3, or f , and that of the latter |,
which is the same as the preceding with the teritis inverted.
It is evident, indeed, that we invert a ratio by inverting the
terms of the fraction, which expresses it, since we make the an-
tecedent take the place of the consequent, and the consequent
that of the antecedent. | or 2 : 3 is the inverse of | or 3 : 2.
The mode of proceeding in such cases may be rendered very
simple ; for we have only to take the numbers denoting the two
sets of workmen, for the quantities of the first kind, and the num-
bers denoting the days, for those of the second, and to place the
one and the other in the order of their magnitude ; proceding
thus we have the following proportion,
Arith. 14
106 ^B Ariihmttk.
15: 27 : : a?: l8,
from which we immediately find x equal to 10.
Recapitulating the remarks already given, we have the fol-
lowing rule; mofce tht number which is of the same kind with
the anspifr the third term^ and the two remaining ones the first
and second, putting the greater j^r the less first, according as tht
third is greater or less than the lerm sought ; then the fimrlh term
will be found by multiply'mg together the second and third, and
dividing the product by the first.
120. 1st. A man placed 3575 dollars at interest at the rate of
5 per cent yearly ; it is asked what will be the interest of this
sum at the end of one year 7
The expression, 5 per cent- interest, means, that for a sum of
one hundred dollars, 5 dollars is allowed at the end of a year ;
if ijben watake the-t^o principals for the quantities of the first
kind, anMi^ in'^ereSt-of those for the second, we shall have,
^^^Sj^^ ^00 : 3575 : : 5 : a?,
a propoi||iqD which^B/ be reduced to 20 : 3575 :: 1 : x, ac-
cording to;;it]|kobsen^n in article 1^5 ; then dividing the two
terms of tt)^Bftfl^HDn by 5, we shall have 4 : 715 : : 1 : x,
whence :r H ^M^^^^w^ ^^ $^ 78,75
•We may also^RHB this question A^ considering that 5 is j\
of 100, aild that '^wl^uently we sh»|[bbtain the interest of any
sum put out at thia^te by taking tJV twentieth part of this sum.
Now ^ of }3575 is }1 78,75 ; a r^plt which agrees with the one
before found. /
2d. iV jnerchant gives his not^'for ^800,00 payable in a year f
the note is s^ to a broker, i^ho advances the money for it 8
months before the time of payment; how much ought the broker
to give ?
As the broker advances, from his own funds, a sum which is
not to be Replaced till the expiration of 8 monthsi it is proper
that he should be allowed interest for his money during thii
time.
Let the interest for a year be 6 per cent, the interest for 8'
months will be y'y, or |, of 6, or 4 ; a sum then of 100 dollars,,
le^t for 8 mwi^s, must be entitled to 4 dollars interest; that is,
Proportion, IGT
he who borrows it ought to return $104. By considering the
$S00^ as a sum so returned for what is advanced by the broker,
we shall have this proportion, 104 : 100 : : 800 : ^, whence we
get $769,231 for the value of x, that is, for the sum the broker
ought to give."**
Quetlionsfor practice.
What is the value of a cwt. of sugar at 5|d. per lb. ?
Ans. 2i. lis* 4d.
What is the value of a chaldron of coals at 11 |d. per bushel?
Ans. II. 14s. 6d.
What is the value of a pipe of wine at lO^d. per pint?
Ans. 44/. 2s.
At 3t. 9s. per. cwf. what is the value of a pack of wool^
weighing 2c wt. 2qrs. 13lb. Ans. 91. 6d. yW
What is the value of 1 |cwt. of coffee at 5|d. per ounce ?
Ans. 61/. 12s.
Bought 3 casks of raisins, each weighing 2cwt. 2qr8. 25lb*
what will they come to at 21. Is. 8d. per cwt. ?j
Ans. 1 11 4|d. tVi-
What is the value of 2qrs. Inl. of velvet at 19s. 8jd. per
English ell? Ans. 8s. lO^d. i^.
Bought 12 pockets of hops, each weighing Icwt. 2qrs. 17lb. ;
what do they come to at 4i. Is. 4d. per. cwt.? "
Ans. 8M.12S. Ijd. j\%.
What is the tax upon 745/. 14s. 8d. at 3s. 6d. in the pound ?
itrp . Ans. 130/. lOs. 0$d. .y,.
f A sum, thus advanced, is called the present $oorih of the sum due
at the expiration of the proposed time.
* The operation by which we find what ought to be given for a
sum of money, when the time of payment is anticipated, belongs to
what is called Discount, There are several ways of cali:ulating
discount, but the above is the most exact, as it has regard merely to
simple interest.
108 Arithmetic.
If I of a yard of velvet cost 7s. Sd. how many yards can I
buy for 13/. 15s. 6d. ? Ans. 28| yaixls.
If an ingot of gold, weighing 9lb. 9oz. 12dwt. be worth 411/.
12s. what is that per grain? Ans. Ifd.
How many quarters of corn can I buy for 1 40 dollars at 4s.
per bushel ? Ans. 26qrs. 2bu.
Bought 4 bales of cloth, each containing 6 pieces, and each
piece 27 yards, at 16/. 4s. per piece, wtiat is the value of the
whole, and the rate per yard ?
Ans. 388/. 16s. at 12s. per yard.
If an ounce of silver be ^orth 5s. 6d. what is the price of a
tankard, that weighs lib. lOoz. lOdwt. 4gr.
Ans. el. 3s. 9|d. j\%.
What is the half year's rent of 647 acres of land at 15s. 6d.
per acre? Ans. 21 U. 19s. 3d.
At $1,75 per week, how many months' board can I have for
100/..^ . j^n*. 47m. 2w.iSir-
Bought 1000 Flemish ells of cloth for 90/. how must I sell it
per ell in Boston to gain 10/. by the whole f Ans. 3s. 4d.
If a gentleman's income is 1750 dollars a year, and he spends
,^1^8. 7d. per day, how much will he have saved at the year's
end? Ans. 167/. 12s. Id.
What is the value of 1 72 pigs of lead, each weighing 3cwt.
2qrs. 17-J^lb. at 8/. 17s. 6d. per fother of 19^cwi f
Ans. 286/. 4s. 4id.
The rents of a whole parish amount to 1750/. and a tax is
granted of 32/. 16s. 6d. what is that in the pound f
V Ans. 4H Tliwy.
If keeping of one horse be 1 l^d. per day, what will be that of
11 horses for a yeari^ Ans. 192/. 7s. 8^d.
A person breaking owes in all 1490/. 5s. lOd. and has in
money, goods, and recoverable debts, 784/. 17s. 4d. if these
things be delivered to bis creditors, what will they get on the
pound ? Am. lOs. e^d. iHM •
What must 40s. pay towards a tax, when 652/. 13s. 4d. is as-
sessed at 83/. 12s. 4d. ? Ans. 5s. lid. UUl-
Bought 3 tuns of oil for 151/. 14s. 85 gallons of which being
Proppnion, 109
damaged, I desire to know how I may sell the remainder per
gallon, so as neither to gain nor lose by the bargain ?
dns. 45. 6-J^d. -gVr.
^ What quantity of water must I add to a pipe of mountain wine,
valued at 33/. to reduce the first cost to 4s. 6d. per gallon ?
Arts, 20f gallons.
If 15 ells of stuff, \ yard wide, cost 37s. 6d. what will 40 ells
of the same stuff cost, being a yard wide ? Ans. 6/. 13. 4d.
Shipped for Barbadoes 500 pairs of stockings at 3s. 6d. per
pair, and 1650 yards of baize at Is. 3d. per yard, and have re-
ceived in return 348 gallons of rum at 6s. 8d. per gallon, and
7501b. of indigo at Is. 4d. per lb. what remains due upon my
adventure i Ans. 24/. 12s. 6d.
If 100 workmen can finish a piece of work in 12 days, how
many arer sufficient to do the same in 3 days ? Ans. 400 men.
How many yards of matting, 2ft. 6io. broad, will cover a floor,
that is 27ft. long, and 20ft. broad. Ans. 72 yards.
How many yards of cloth, 3qrs. wide, are equal in measure to
30 yards 6qrs. wide i* Ans. 50 yards.
A borrowed of his friend B 250/. for 7 months, promising to
do him the like kindness ; sometime after B had occasion for
300/. how long may he keep it to receive full amends for the
favor f Ans- 5 months and 25 days.
If, when the price of a bushel of wheat is 6s. 3d. the penny
loaf weigh Ooz. what ought it to weigh when wheat is at Ss. 2^d.
per bushel ? Ans. 6oz. 13dr.
If 4^cwt. can be carcied 36 miles for 35 shillings, how many
pounds can be carried 20 miles for the same money ?
Ans. 9071b. t^.
How many yards of canvass, that is an ell wide, will line 20
yards of say, that is 3qrs wide } Ans. 12yds. ^
If 30 men can perform a piece of work in 1 1 days, how many
men will accomplish another piece of work, 4 times as large, in a
fifth part of the time ? Ans. 600.
A wall that is to be built to the height of 27 feet, was raised 9
feet by 12 men in 6 days ; how many men must be employed to
finish the wall in 4 days at the same rate of working ?
Ans. 36.
lit) , Arithmetic
Tffoz. cost ^ll* what will loz. cost ? Ans. \U 5s* 8d.
If ^ of a ship cost 273/. 2s. 6d. what is -3^ of her worth ?
Am. 2211. t2s. Id.
. -At 1^2. per cwt. what does S^lb. come to f An$» lOf d.
If I of a gallon cost \l. what will 7 of a tun cost. Ans. 1401.
A person having f of a coal mine, sells \ of his share for
llXh what is the whole mine worth i Ans. 2&0L
If, when the days are 13|- hours long, a traveller perform his
journey in 35-}^ days ; in how many days will he perform the
same journey, when the days are 11-^ hours long ?
Am. 4afi} days.
A regiment of soldiers, consisting of 976 men, are to be new
clothed, each coat to contain 2^ yards of cloth, that is Ifyd.
wide, and to be lined with shalloon, lyd. wide ; how many yards
of shalloon will line them ? . Ans. 4531yds. Iqr. 2-f nl.
Compound Proportion.
121. Proportion is also applied to questions, in which the re-
lation of the quantity required, to the given quantity of the same
kind, depends upon several circumstances, combined together ; it
is then called Compound Proportion^ or Double Rule of TTiree.
See some examples.
It is required to find how many leagues a person would go
in 1 7 days, travelling 10 hours in a day, when he is known to have
travelled 1 1 2 leagues, in 29 days, employing only 7 hours a day.
This question may be resolved in two ways, we will first give
the one that leads to Compound Proportion.
In each case, the number of leagues passed over depends upon
two circumstances, namely, the number of days the man travels,
and the number of hours he travels in each day.
We will not at first consider this latter circumstance, but sup*
pose the number of hours be the same in each case ; the ques-
tion then will be ; a person in 29 days- travels 1 1 2 leagues, horn
9nany will he travel in 17 days f This will furnish the following
proportion ;
39: 17 :: 112:x.
Compound Proportttm* 111
The fourth term will be equal to 112 multiplied by 17 and di-
vided by 29, or -^S-*- leagues.
Now, to take into consideration the number of hours, we must
say, if a person travelling 7 hours a day, for a certain number of
days, has travelled -^f?^ leagues, how far will he go in the same,
time, if be travel 10 hours a day f This will lead to the follow-
ing proportion,
b. h. ^ L
7:10:: ^^ : c^,
which gives for the fourth term, or answer, 93,793 leagues
nearly.
The question may also be resolved by observing, that 59 days^
travelling at 7 hours a daty, is equal to 203 hours' travelling ;
and that 17 days, at 10 hours a day, amounts to 170 hours; the
problem then is reduced to this proportion,
203 : 170 :: 112 : ar,
by which we find the distance he ought to travel in 170 hours,
according to what he performed in 203 hours.
We see by the first mode of resolving the question, that 112
leagues has to the fourth term or answer, the same proportion,
that 29 days has to 17, and that 7 hours has to 10; stating this
in the form of a proportion, we have
d.
29
d. d. j\
h > : : 112 : a?,
7 :10)
by which it appears, that 1 1 2 is to be multiplied by both 1 7 and
10, and to be divided by both 29 and 7 ; that is, 1 12 is to be mul-
tiplied by the product of 17 by 10, and divided by the product
af 29 by 7, which is the same as the second method of resolving
the question.
192. Again, if 9 labourers^ working 8 hours a day, have
spent 24 days in digging a ditch 65 yards long, 13 wide, and 5
deep, how many days will it take 71 labourers of equal ability,
working 11 hours a day, to dig a ditch 327 yards long, IS
broad, and 7 deep?
Here is a question very complicated in appearance, but which
inay be resolved by proportion.
If all the condition of these two cases were alike, except the
2
Arithmetic.
nber of men and the number of days, the question would con*
: only in finding in how many days 71 men would perform the
rk, which 9 men have done in 24 days ; we should have then,
71 : 9 : : 24 : x,
: instead of calculating the number of days, we will only indi-
e as in article 82, the numbers to be multiplied together, and
ce as a denominator those by jvhich they are to be divided ;
thus have for x days,
24 by 9
71 *
I as the first labourers worked only 8 hours a day, while the
ers worked 1 1, the number of days required by the latter will
less in proportion, which will give
, , ■ 24 by 9
11:8:: i — :x;
71
ence we conclude that the number of days, in this case, is
24 by 9 by 8
71 by 11
rhis number is that of the days necessary for 71 labourers,
rking 1 1 hours a day to dig the first ditch.
rhe ditches being of unequal length, as many more days will
lecessary, as the second is longer than the first, thus we shall
e
/;c Qo^ 24 by 9 by 8
65 : 327 : : -, — -^J — : a:,
71 by 11
I the number of days, this new circumstance being consider-
will be
24 by 9 b y 8 by 327
71 by 11 by 65
raking into consideration the breadths, which are not alike,
have
,o.,g.. 24by9by8by327
^^ • ^^ • ' 71 by 11 by 65 ' ""'
I, consequently, the number of days required changes t<^
24 by 9 by 8 by 327 by 18
71 by 11 by 65 by 13
Compound Proportion. 113
liastly, the depths being different, we have
^,^,, 24by9by8by327byl8
71 by 11 by 65 by 13 * '
dnd the number of dajs, resulting from the concurrence of all
these circumstances, is
24 by 9 by 8 by 327 by 18 by 7
71 by 11 by 65 by 13 by 6 *
Performing the multiplications and divisions, we get the answer
required, 21 days ^f ||«|}.
123. This number is equal to 24 multiplied by the fracti<mal
quantity
9by 8by 327by 1 8 by 7
71 by 11 by 65 by 13 by 5'
but this last quantity, which expresses the relation of the num-
ber of days given, to the number of days required, is itself the
product of the following fractions ;
• • 397 18 1
Now going back to the denominations given to these numbers in
the statement of the question, we see that these fractions are the
ratios of the men and the hours, of the lengths, the breadths, and
the depths of the two ditches ; hence it follows, that the ratio of
the number of days given, to the number of days sought, is equal
to the product of all the ratios, which result from a comparison
of the terms relating to each circumstance of the question. •
This may be resolved in a simple manner by first finding the
value of each of these ratios ; for, by multiplying together the
fractions that express them, we form a fraction which represents
the ratio of the quantity required to the given quantity of the
same kind.
This fraction, which will be the product of all the ratios in the
question, will have for its numerator the product of all the ante-
cedents, and for its denominator, that of all the consequents. A
ratio resulting, in this manner, from the multiplication of sever-
al others, is called a compound ratio.
By means of the fractional expression
9 by 8 by 327 by 18 by 7
71byllby65by 13by 5'
Ariih. 15
1 14 Arithmetic.
and the given number of days, 34, we make the following pro-
portion ;
71 by il by 65 by 13 by 5 : 9 by 8by 327by 18 by 7:: 24:ar,
which may also be represented in this manner, as was the pre-
ceding example ;
24 IX.
71
: 9")
11 :
8
65 :
;327 \
13;
'. 18
5:
• 7J
Our remarks maybe summed up in this rule; Make the number
which is of the same kind vnth the required answer^ the third term ;
and of the remaining numbers^ take any two that are of the same
kind^ and place one for a first term and the other for a second term
according to the directions in simple pr^ortion ; then any other two
if the same klnd^ and so on^ till all are used ; lastly ^ multiply the
third term by the product of the second terms^ and divide the result
by the product of the first terms^ and the quotient mil be the fourth
ierm^ or -answer required*
Examples for practice.
If 100/. in one year gain 5/. interest, what will be the interest
of 750/. for 7 years ? Ans. 262/. 10s.
What principal will gain 262/. 10s. in 7 years, at 5/. per cent,
per annum ? Ans, 750/.
If a footman travel 130 miles in 3 days, when the days ate 12
hours long; in how many days, of 10 hours each, may he travel
360 miles ? ^ Ans. 9J|
If 120 bushels of co^tican serve 14 horses 56 days ; how many
days will 94 bushels s^ve 6 horses t Ans. 102^| days.
If 7oz. 5dwt. of bread be bought at 4f d. when corn is at 4s.
2d. per bushel, what weight of it may be bought for Is. 2d. when
the price per bushel is 5s. 6d.? Ans. lib. 4oz. 3|^fdwt.
If the transportation of 13cwt. IqK 72 miles be 2/. 10s. 6d.
what will be the transportation of 7cwt. Sqrs. 112 miles?
Ans. 21. 5s. lid. IrYiFq*
A wall, to be built to the height of 27 feet, was raised to the
height of 9 feet by 12 men in 6 days ; how many men must be
employed to finish the wall in 4 days at the same rate of work-
ing ? Ans. 36 men.
c
Fellowship* 115
If a regiment of soldiers, consisting of 939 men, consume 351
quarters of wheat in 7 months ; how many soldiers will consume
1464 quarters in 5 months, at that rate? Ans. 5483rW.
If 248 men, in 5 days of 11 hours each, dig a trench 230
yards long, 3 wide, and 2 deep ; in how many days of 9 hours
in length, will 24 men dig a trench of 420 yards long, 5 wide,
and 3 deep? Ans. 2887'/,.
Fellowship.
124. The object of this rule is to divide a number into parts,
which shall have a given relation to each other ; we shall see in
the following example its origin, and whence it has its name.
Three merchants formed a company for the purpose of trade;
the first advanced 25000 dollars, the second 18000, and the third
42000 ; after some time they separated, and wished to divide the
joint profit, which amounted to 57225 dollars ; how much ought
each one to have ?
To resolve this question we roust consider, that each man's
gain ought to have the same relation to the whole gain, as the
money he advanced has to the whole sum advanced ; for he who
furnishes a half or third of this sum, ought plainly to have a
half or third of the profit. ^ In the present example, the whole
sum being 85000 dollars, the particular sums will be respec-
tively UUh iiiU^ mn of it; and if we multiply these
fractions by the whole gain, 57225, we shall obtain the gain be-
longing to each man. It is moreover evident, that the sum of
the parts will be equal to the whole gain, because the sum of the
above fractions, having its numerator equal to its denominator,
is necessarily an unit.
We have therefore these propositions ;
$ $ i
85000 : 25000 : : 57225 : to the first man's gain,
85000 : 18000 : : 57225 : to the second man's gain,
85000 : 42000 : : 57225 : to the third man's gain,
which may be enunciated thus ;
The whole sum advanced : to each man's particular sum : : the
whole gain : to each man's particular gain.
1 1 6 Arithmttic.
Bj simplifying the first ratio of each of these proportions we
have
%
85 : 26 : : 57225 : to the gain of the \^ or $16830J{,
85 : 18 : : 57225 : to the gain of the 2<i- or $12118|J,
85 : 42 : : 57225 : to the gain of the S^-'or 428275^^.
If all the sums advanced had been equal, the operation would
have been reduced to dividing the whole gain bj the number of
sums advanced ; we may reduce the question to this in the
present case by supposing the whole sum, $85000, divided into
85 partial sums, or stocks of $1000 each; the gain of each of
these sums will evidently be the 85*^ part of the whole gain ; and
nothing remains to be done, except to multiply this part severally
by 25, 18, and 42, considering the sums 25000, 18000, and 42000
as the amounts of 25 shares, 18 shares, and 42 shares.
In commercial language the money advanced is called the
capital or stocky and the gain to be divided, the dividend*
The following quesriou is very similar to that just resolved.
125. It is required to divide an estate of 67250 dollars among
3 heirs, in such a manner, ihat the share of the second shall be
I of that of the first, and the share of the third | of that of the
second.
It is plain that the share of the third, compared with that of the
first, will be } of f of it, or ^\ ; then the three parts required
will be to each other in the proportion of the numbers 1, f and
/,. Reducing these to a common denominator, we find them
If, Z^, and y\, and have the three numbers 20, 8, and 7, which
arc proportional to the first ; but as their sum is 35, it is plain,
that if we take three parts expressed by the fractions fj, /y,
3"<1 /t9 ^cy w*^l ^^ ^" ^^^ required proportion. The question
will then be resolved by taking |{, then /7,and then -^-g of 67250
dollars, which will give the sums due to the heirs, according to
the manner prescribed, namely ;
$3842811, $15371|i, and $13450.
126. Again, lere are two fountains, the first of which will
fill a certain reservoir In 2^ hours, and the second will fill the
same reservoir in 3| hours ; how much time will be required to
Fellowship. 117
£11 the reservoir, hj means of both fountains running at the same
time?
We must first ascertain what part of the reservoir will be filled
by the first fountain in any given time, an hour for instance. It
is evident that, if we take the content of the reservoir for unity,
we have only to divide 1 by 2|, or f , which gives us f for the
part filled in one hour by the first fountain. In the same man-
ner, dividing 1 by 3|, or y, we obtain xV ^^^ the part of the
reservoir filled in an hour by the second fountain ; consequently,
the two fountains, flowing together for an hour, will fill | added
to y*j, or II of the reservoir. * If we now divide 1, or the con-
tent of the reservoir, by ||, we shall obtain the number of
hours necessary for filling it at this rate ; and shall find it to be
11 or an hour and an half*
Authors who have written upon arithmetic, have multiplied and
varied these questions in many ways, and have reduced to rules
the processes which serve to resolve them; but this multiplica-
tion of precepts is, at least, useless, because a question of the
kind we have been considering may always be solved Vith facil-
ity by one who perceives what follows from the enunciation,
especially when he can avail himself of the aid of algebra ; we
shall therefore proceed to another subject*
127. Besides the proportions composed of four numbers, one of
the two first of which contains the other as many times as the
corresponding one of the two last contains the other ; it has been
usual to consider as such the assemblage of four numbers, such
as 2, 7, 9, 14, of which one of the two first exceeds the other as
much as the corresponding one of the two last exceeds the other.
These numbers, which may be called equidifftrent^ possess a
remarkable property, analogous to that of proportion ; for the sum
of the extreme terms, 2 and 14, is equal to the sum of the means
7 and 9.*
* The ancients kept the theory of proportions very distinct from
the operations of arithmetic Euclid gives this theory in the fifth
book of his elements, and as he applies the proportions to lines^ it \%
apparent, that we thence derive the name o( geometrical proportion ;
US Arithmetic*
To show this property to be general, we mast observe, that the
second terra is equal to the first increased by the difference, and
that the fourth is equal to the third increased by the difference;
hence it follows, that the sum of the extremes, composed of the
first and fourth terms, must be equal to the first increased by the
third increased by the difference. Also, that the sum of the
means, composed of the second and third terms, must be equal
to the first increased by the difference increased by the third
term; these two sums, being composed of the same parts, must
consequently be equal. •
We have gone on the supposition, that the second and fourth
terms were greater than the first and third ; but the con-
trary may be the case, as in the four numbers 8, 5, 15, 12; the
second term will be equal to the first decreased by the difference,
and the fourth will be equal to the third decreased by the differ-
ence. By changing the Mord increased into decreased^ in the
preceding reasoning, it will be proved that, in the present
case, the sum of the extremes is equal to that of the means.
We shall not pursue this theory of equidifferent numbers fur-
ther, because, at present, it can be of no use.
Questions for practice.
A and B have gained by trading }182. A put into stdck
(300 and B $400 ; what is each person's share of the profit 1
Ans. A (78 and B (104.
and that the name of arithmetical proportion was given to an assem*
binge of equidifferent numbers, which were not treated of till a much
later period. These names are very exceptionable ; the word propot^
tion has a determinate meaning, which is not at all applicable to
equidifferent numbers. Besides, the proportion called geometrical it
not less arithmetical than that which exclusiYely possesses the latter
name. M. Lagrange, in his Lectures at the llformal school, has pro-
posed a more correct phraseology, and I have thought proper to
follow his example.
Equidifferenccj or the assemblage of four equidifferent numbers,
or arithmetical proportion, is written thus ; 2 . 7 s 9 . 14.
Among English writers the following form is used ;
2 . . 7 : : 9 . . 14.
Fdlomship, 119
Divide {120 between three persons, so that their shares shall
be to each other as 1, 2, and 3, respectively.
Ans. j(20, $40, and 60.
Three persons make a joint stock* A puts in }1 85,66, B
$93,50, and C $76,85 ; they trade and gain $222 ; what is each
person's share of the gain ?
Ans. A $114,17^V't\ti B $60,57^/^3^, and C 47i25HHf-
Three^merchants, A, B, and C, freight a ship with 340 tuns of
wine; A loaded 110 tuns, B 97, and C the rest. In a storm the
seamen were obliged to throw 85 tuns overboard ; how much
must each sustain of the loss ?
Ans. A 27i, B 24|, and C 33{.
A ship worth $860 being entirely lost, of which \ belonged to
A, \ to B, and the rest to C ; what loss will each sustain, sup-
posing $500 of her to be insured ?
Ans. A $45, B $90, and C $225.
A bankrupt is indebted to A $277,33, to B $305,17, to C
$152, and to D $105. His estate is worth only $677,50; how
must it be divided ? Ans. A $223,8 If f| J, B $246,28tV/j9
C $1 22,66f f II, and D $84,73Hf f •
A and B, venturing equal sums of money, clear by joint trade
$154. By agreement A was to have 8 per cent, because he
spent his time in the execution of the project, and B was to have
only 5 per cent. ; what was A allowed for his trouble ?
Ans. 35,5311.
Three graziers hired a piece of land for $60,50. A put in 5
sheep for A\ months, B put in 8 for 5 months, and C put in 9
for 6| months ; how much must each pay of the rent ?
Ans. A $U,25, B $20, and C $29,25.
Two merchants enter into partnership for 18 months; A put
into stock at first $200, and at the end of 8 months he put in
$100 more ; B put in at first $550, and at the end of 4 months
took out $140. Now at the expiration of the time they find they
have gained $526 ; what is each man's just share ?
Ans. A's $192,95Tl}y,^'3 $333,04|i|}.
A, with a capital of $1000, began trade fanu:^ 1, 1776, nnd
meeting with success in business he took in B 4pnrtDcr, wiih a
capital of $1500 on the first of March following. Three months
ISO Arilhmettc.
after that thej sdmit C as a third partner, who brought into
the stock ^2800 ; and after trading together till the first of the
next year, they find the gain, since A commenced business, to be
}1 776,50. How must this be divided among the partners?
Ans. A's $457,4«||f
B's 571,83fff
C's 747,1 9iH.
Alligation.
1 28. We shall not omit the rule of alligation, the object of
which is to find the mean value of jseveral things of the same
kind, of several values ; the follow^xamples will suflSciently
illustrate it.
A wine merchant bought several kinds of wine, namely ;
130 bottles which cost him 10 cents each,
75 at 15
231 at 12
27 at 20
he afterwards mixed them together ; it is required to ascertain
the cost of a bottle of the mixture. It will be easily perceived,
that we have only to find the whole cost of the mixture and the
number of bottles it contains, and then to divide the first sum
by the second, to obtain the price required.
Now, the 130 bottles at 10 cents cost 1300 cents
75 at 15 cost 1125,
231 at 12 cost 2772,
27 at 20 cost 540,
then 463 bottles cost 5737 cents.
5737 divided by 463 gives 12,39 cents, the price of a bottle of
the mixture.
The preceding rule is also used for finding a mean of difier-
ent results, given by experiment or observation, which do not
agree with each other. If for instance, it were required to
know the distance between two points considerably removed
from each other, and it had been measuj^d ; whatever care
might have been used in doing this, there would always be a
AUigalian* 131
little uncertainty in the result, on account of the errors inevita-
bly committed by the manner of placing the measures one after
the other.
We will suppose that the operation has been repeated several
times, in order to obtain the distance exactly, and that twice it
has been found 3794yds. 2ft., that three other measurements have
given 3795yds. 1ft., and a third 3793yds. As these numbers are
not alike, it is evident that sum tnust be wrong, and perhaps all.
To obtain the means of diminishing the error, we reason thus ;
if the true distance had been obtained by each measurement, the
sum of the results would be equal to six times that distance, and
it is plain that this would also be the case, if some of the results
obtained were too little, and others too great, so that the increase,
produced by the addition of the excesses, should make up for
what the less measui*emenls wanted of the true value. We
should then, in this last case, obtain the true value by dividing
the sum of the results by the number of them.
This case is too peculiar to occur frequently, but it almost al-
ways happens, that the errors on one side destroy a part of those
on the other, and the remainder, being equally divided among
the results, becomes smaller according as the number of results
is greater.
According to these considerations we shall proceed as follows ;
yds. ft. ft
We take twice 3794 2 or 7589 1
yds. ft,
3 times 3795 1 or 11386
yds.
once 3793 or 3793
6 results, giving in all 22768 1.
Dividing 22768yds. 1ft. by 6, we find the mean value of the
required distance to be 3794yds. 2ft.
129. Questions sometimes occur, which are to be solved by a
method, the reverse to that above given. It may be required,
for example, to find what quantity of two different ingredients it
would take to make a mixture of a certain value. It is evident,
that if the value of the mixture required exceeds that of one of
the ingredients just as much as it falls short of that of the other,
we should take equal quantities of each to make the compound*
AriA. 16
12S Arithmetic
c*
So also, if the value of th& mixture exceeded that of one twiee as
much as it fell short of that of the other, the proportion of the
ingredients would be as one half to one. To mix wine at }2 per
gallon with wine at }3, so that the compound shall be worth
$2^50^ we should take equal quantities, or quantities in the
proportion of 1 to 1. If the mixture were required to be worth
(2,66|, the quantities would be in the proportion of ^ to 1, or of
I 1
r^ to -— ; and, generally, the nearer the mixture rate is to
that of one of the ingredients, the greater must be the quantity of
this ingredient with respect to the other, and the reverse ; hence^
To find the ppoportitm of two ir^redierUs of a given value^ neces*
sary to constitute a compound of a required valuey make the differ^
ence between the value of each ingredient and that of the con^pound
the denominator of a fraction^ whose numerator is ofu^ and these
fractions tdHI express the proportion required; and being reduced
to a common denomhiator, the numerators will express the same
proportion, or show what quantity of each ingredient is to be
taken to make the required compound.
When the compound is limited to a certain quantity, the pro-
portion of the ingredients, corresponding to it, may be found by
saying ; as the whole quantity, found as above, is to the quantity
required, so is each part, as obtained by the rule, to the requu-ed
quantity of each.
Let it be required, for example, to mix wine at 5s. per gallon
and 8s. per gallon, in such quHfitities that there may be 60 gal-
lons worth 6s. per gallon. The difference between' 68. and 5s«
is 1, and between 6s. 'and 8s. is 2, giving for the required quan-
tities the ratio of | to ^, or 2 to 1 ; thus, talking x equal to the
quantity at 5s% and y equal to the quantity at Bs. we have these
proportions ; 3 : 60 : : 2 : «, and 3 : 60 : : 1 : y, giving, for the
answer, 40 gallons at 5s. and 20 gallons at 8s. per^Uon.
Also, when one of the ingredients is limited, we may say ; as
the quantity of the ingredient found as abovie, is to the required
quantity of the. same, so is the qusntity of the other ingredient
to the proportional part required.
For example, I would know how many gallons of water at
Os. per gallon, 1 must mix with thirty gallons of wine at ^s. per
AUigaiton. 133
gallon, so that the compound may be worth 5s. per gallon. First,
the difference between Os. and 5s« is d ; and the difference be-
tween 6s« and 5s« is 1 ; the quantity of water therefore will be to
that of the wine, as j to |, or as 1 to 5. Then, from this ratio,
we institute the proportion, 5 : 30 : : 1 : a?, which gives 6, for the
number of gallons required.
As we have found the proportion of two ingredients necessary
to form a compound of a required value, so also we may con-
.sider either of these in connexion with a third, with a fourth,
and so on, thus making a compound of any required value, con-
sisting of any number whatever of simple ingredients. The two
ingredients used, however, must always be, one of a greater and
the other of a less value, than that of the compound required.
A grocer would mix teas at 12s. and 10s. with 401b. at 4s. per
pound, in such proportions that the composition shall be worth Bs.
per lb. If he mix only two kinds, the one at 4s. and the other at
10s, their quantities will be in the ratio of | to ^, or 1 : 2 ; and
if he mix the tea at 4s. also with that at 12s. their ratio will be
that of { to j, or of 1 to 1. Adding together the proportions of
the ingredient, which is taken with each of the others, we find
the several quantities, at 4s. 10s. and 12s. to be as 2, 2, and 1.
And taking x for the number of lbs. at 10s. and y for the quan-
tity at 12s. we have the following proportions ;
2 : 40 : : 2 : 0? ; and 2 : 40 : : 1 : y ;
giving, for the answer, 40lb. at 10s. and 201b. at 12s. per pound.
The problems of the two last articles are generally distin-
guished by the names of alligation medial^ and alligation alter'
note. A full explanation of the latter belongs properly to algebra.
Examples.
A composition being made of 51b. of tea at 7s. per pound, 91b.
at 8s. 6d. per pound, and H^Ib. at 5s. lOd. per pound ; what is
a pound of it worth ? ' Ans. 6s. lOj-d.
How much gold, of 15, of 17, and of 22 carats* fine, must be
mixed with 5oz. of 18 carats fine, so that the composition may be
90 carats fine ? Ans. 5oz. of 15 carats fine, 5 of 1 7, and 25 of 22.
t A carat is a twenty-fourth part ; 22 carats fine means f| of pure
metal. A carat is also divided into four parts, called grains of a carat.
\
\
124 Jiriihmttk.
I
/
^
Miscellaneous Questions for practice.
What number, added to the thirty-first part of 3813, will make
he sum 200? * -< ^^ Ans. 11.
The remainder of a division is 325, the quotient 467, and the
livisor is 43 more than the sum of both ; what is the dividend f
Ans. 390270.
Two persons depart from the same place at the same time ;
ie one travels 30, the other 35 miles a day ; how far are they
istant at the end of 7 days, if they travel both the same road ;
nd how far, if they travel in contrary directions f
Ans. 35, and 455 miles.
A tradesman increased his estate annually by 100/. more than
part of it, and at the ^end of 4 ye^j^found that his estate
mounted to 1-03427. js. ffi^ What Hf^e at first ?
Ans. 4000/.
Divide 1200 acres of land among A, B, and C, so that B may
ave 100 more than A, and C 64 more than B.
Jhis. A 312, B 412, and C 476.
Divide 1000 crowns ; give A 120 more, and B 95 less, than C.
Ans. 445, B 230, C 325.
What sum of money will amount to 132/. 16s. 3d. in 15 months,
t 5 per cent, per annum, simple interest ? Ans. 125/.
A father divided his fortune among his sons, giving A 4 as
ften as B 3, and C 5 as often as B 6 ; what was the whole
igacy, supposing A^s share 5000/. ? Ans. 11875/.
If 1000 men, besieged in a town with provisions for 5 weeks,
ich man being allowed 16oz. a day, were reinforced with 500
len more ; on hearing, that they cannot be relieved till the end
r 8 weeks, how many ounces a day must each man have, that
le provision may last that time f Ans. 6f • |
What number is that, to which if f of f be added, the sum ;
ill be 1 } Ans. f|.
A father dying left his son a fortune, \ of which he spent in 8 «*
lonths ; -f of the remainder lasted him twelve months longer ;
fter which he had only 410/. left. What did his father bequeath
ira? • w3n5. 966/. 13s. 4d.
MiscelUtneaus ^estions* 125
*
A guardian paid his ward 3500Z. for 2500Z. which he had in
his hands 8 years. What rate of interest did he allow him ?
Ans. 5 per cent«
A person, being asked the hour of the day, said, the time past
noon is equal to 7 of the time till midnight. What was the time f
Ans. 20min. past 5.
A person looking on hid watch, was asked, what was the time
of the •day ; he answered, it is between 4 and 5 ; but ^a more
partfcular answer being required, he said, that the hour and
minute hands were then exactly together. What was the time ?
Ans. 21-^ min. past 4.
With 12 gallons of Canary, at 6s. 4d. a gallon, I mixed 18
gallons of white wine, at 4s. lOd. a gallon, and 12 gallons, of
cider, at 3s. Id. a gallon. At what rate must I sell a quart of
this composition, so as to clear 10 per cent, f Ans, Is. 3yd.
What length must be cut off a board, Bf inches broad, to con*
* tain a square foot ; or as much as 12 inches in length and 12 in
breadth? Ans. IT^fin.
What dijSerence is there between the interest of 350Z. at 4 per
cent, for 8 years, and the discbunt of the same sum,vat the same
rate and for the san^e time ? Ans. 271. S^s.
A father devised iV of his estate to one of his sons, and iV of
the residue to another, and the surplus to his relict for life ; the
children's legacies were^found to be 2571. 3s. 4d. different. What
money did he leave for the widow ? Ans. 6351. lOffd.
What number is that, from which if you take ^ of f , and to the
remainder add ^ of -^y the^um will be 10 ? Ans. IOq^^At
A man dying left his wife in expectation that a child would
be afterward added to the surviving family ; and, making hin^
will, ordered, that if the child were a son, 1 of his estate should
belong to him, and the remainder to his mother ; but if it were
a diughter, he appointed the mother f, and the child the remain-
der. But it happened, that the addition was both a son and a
daughter, by which the mother lost in equity 2400Z. more than
if it had been only a daughter. What would have been her
dowry, had she had only a son ? Ans. 2100/.
156 JriAnutic. ^
' A young bare starts 40 rods before a greyhound, and ' is not
perceived by him till she has been up 40 seconds ; she scuds
away at the rate of 10 miles an hour, and the dog, on view,
makes after her at the rate of 18. How long will the course
continue, and what will be the length of it from the place, where
the dog set out f Jlns. 60^ seconds, and 535 yards run.
A reservoir for water has two cocks to supply it ; by the first
alone iunay be fiUed in 40 minutes, by the second in 50 minutesi
and it has a discharging cock, by which it may, when full, be
emptied in 95 minutes. Now these three cooks being all left
open, the influx and efflux of the water being always at the same
rate, in what time would the cistern be filled f
Afu. 3 hours 20 mintues.
A sets out from London for Lincoln precisely at the same time,
when B at Liocoln sets out for London, distant 100 miles ; after 7
hours they met on the road, and it then appeared, that A bad
ridden l-J- mile an hour more than B« At what rate an hour
did each of them travel? AnsJ^A 7|^f, B 6^ miles.
Whft part of 3 pence is a third part of 2 pence f An9' !•
* A has by him l^cwt. of tea, the prime cost of which was 96L
sterling. Now interest being at 5 per cent, it is required to find
how h^ must rate it per poqnd to B, so ttiat by taking his nego-
tiable note, payable at 3 months, he may clear 90 guineas by the
bargain ? Ana. 14s. li|d- sterling.
There > is an island 73 miles in circumference, and 3 footmen
aU start together to travel the same way about it ; A goes 5
miles a day, B 8, and C 10; when will they all come together
again f An$. 73 days.
A man being asked how many sheep he had in his drove, said,
if he had as many more, half as many more, and 7 sheep and a
half, he should have 20; how many had he ? Am. 5.
A person left 40s. to 4 poor widows, A, B, C, and D ; in A
he left ^, to B i, to C i, and to D i*, desiring the whole might
be distributed accordingly ; what is the proper share of each ?
Ans. A's share 14s. Ud. B's 10s. ^d. C'a 8s. 5^. D's
7s. ^nrd* •
ABteeUaneaui Questions. 1%1
A general, disposing of his army into a square, finds he has
364 soldiers over and above ; but increasing each side with one
soldier, he wants 25 to fill up the square ; how many soldiers
had he? Ans. 24000.
There is a prize of 2122. 14s. 7d. to be divided among a cap-
tain, 4 men, and a boy ; the captain is to have a share and a
half; the men each a share; and the boy -)- of a share ; what
ought each person to have ?
wSnV The captain 542. 148. fd. each man 36/. 9s. 4^d. and the
boy 122. 3s. l|d. ^
A cistern, containing 60 gallons of water, has 3 unequal cocks
for discharging it ; the greatest cock will empty it in one hour,
the second in 2 hours, and the third in 3 ; in what time will it be
emptied, if they all run together ? Ans. 2>2Yrmm\xies.
In an orchard of firuit trees, ^ of them bear apples, \ pears, \
plumbs, and 50 of them cherries : how many trees are there in
all.^ Ans. 600.
A can do a piece of work alone in 10 days, and B in 13 ; if
both be set about it together, in what time will it be finished f
Ans. dyf days.
A, B, and C are to share 1000002. in the proportion of *|^, -},
and -I*, respectively ; but C's part being loet by his death, it is
required to divide the whole sum properly between the other two*
Ans. A's part is 57142^2. and B's 42857-f2.
T
V.'. ■^■>.
\
NOTE.
Summation of a continued Fraction.
«
^ When in the course of a calculation we meet with a fraction
whose numerator and denominator are pretty large, and have
DO common factor, we seek approximate values of this fraction,
which are expressed by more simple numbers, with a view to
forming a more exact idea of it.
If we have, for example, the fractional number V^V , we obtain,
at first, the whole number, and there results 1 and flf. Now, to
form a more simple idea of the fraction |}f , we endeavour to
compare it with a part of unity, in order that we may have only.
one term to be inquired into, and for this purpose we divide the
two terms by 316 ; we find 1 for the quotient of the numerator,
and 4/|-\ for that of the denominator ; this last quotient, being
between 4 and 5, shows that the fraction fjf is between i and |.
By stopping at this point, we see that the second approximate
value of the expression VVy is 1 and |, or |. But this value is
too great, for the true value would be equal to 1 plus 1 divided
by 4 added to /i-\, which is written thus; 1 — L..
4/rV
To form an exact idea of the expression 1 — i — i it is necessa-
4M
ry to consider it as indicating the quotient of the whole nmnber
1 divided by the whole number 4 accompanied by the firaction
9 3
9
If vr6 divide the two terms of /tt ^7 ^^ ^^^ quotient will be
L- ; neglecting the -fj which accompany the whole number 9,
there will be ^ only instead of /|\, and consequently, l-±., will
be a third approximate value of VVV? ^ value which will be too
small, since 9 being less than the true quotient of 216 by 23,
the fraction i will be greater than that which ought to accom-
pany 4, and consequently the divisor 4| will be greater than the
exact divisor 4/fV, and the quotient -L. smaller than the true
quotient. ^
Arith. 17
130 Note.
By reducing the whole number 4 with the fraction which
accompanies it, and performing the division according to the
method of art. 74, we obtain /^ ; and we have 1 and /y or |f
for the third approximate vulue of VVV-
The exact expression of this value being 1 — L, ,
9 9
if we divide the two terms of A by 9, we shall have 1 -i ;
4_J_
9-1
neglecting the fraction f , there will remain 1 — ^i
94
a value too great; for the fraction ^ being greater than — L.i
whose place it occupies, will form, by being joined with 9, a de-
itOminator too great; the fraction joined to -4 will consequently
be too small, and the denominator being too small, the fraction
itself will be too great.
By reducing, at first, 9^ to a fraction, we have y ; -1-* will
then be Z^, and the approximate value will become 1_! — ; now
_i- gives If, which joined to unity becomes l^f, or f } for the
fourth approximate value of VVV-
Resuming the expression, 1 — I ,
2f
we divide the two terms of the last fraction f by 5, and obtain
l.JL_,and thence 1— i ,
- i ' 4—:
^ 9-J
2—2
1 *
1 *
T
neglecting the fraction |, there will remain 1—
4.
9— L
and it will be seen as before, that this value is less than the true
value.
The fraction -i- reduces itself to \] and since the preceding
Note. 131
-L- gives ^V) ^^^ ^^^^ preceding becomes -L. equal to -f^^ ; so
that the fifth approximate value is \^^j or |f|.
Dividing, finally, bj 4 the two terms of the fraction | which
was neglected above, we have for a quotient ^J— and by sup-
pressing the fraction |, we obtain the new value 1— J
2.
It
greater than the true value. If we reduce, successively, all the
denominators to a fraction, to obtain the simple fraction which
it represents, we shall find lyVs ^^ fH- ^y restoring the
fraction \ to the side of the last denominator, we form the ex«
pression 1 — \ ,
1
U
which being reduced like the preceding, reproduces the fraction-
al number VtV-
We may pursue the same process with any other fraction,
and obtain a series of approximate values, alternately greater
and less than the true value, if it is a fraction properly so called,
or alternately less and greater, if, as in the preceding example,
the numerator exceeds the denominator.
The developments above found for the expression VtV ^^^
called continued fractions^ which may be defined in general
thus : — Fractions whose denominator is composed of a whole number
and a fraction^ which fraction has for a denominator also a whole
number and a fraction^ ^cJ^^
APPENDIX,
CONTAINING TABL^ OF VARIOUS WEIGHTS AND MEASURES.
A*et9 French Weights and Meoiures.
The weights and measures in common use are liable to great
uncertainty and inconvenience. There being no fixed standard at
hand, by which their accuracy can be ascertained, a great variety
of measures, bearing the same name, has obtained in different
countries. The foot, for instance, is used to stand for about a
hundred different established lengths. The several denomina-
tions of weights and measures are also arbitrary, and occasion
most of the trouble and perplexity, that learners meet with in
mercantile arithmetic.
To remedy these evils, the French government adopted a
new system of weights and measures, the several denomina-
tions of which proceed in a decimal ratio, and all referrible to a
common permanent standard, established by nature, and acces-
sible at all places on the earth. This standard is a meridian of
the earth, which it was thought expedient to divide into 40 mil-
lion parts. One of these parts is assumed as the unit of length,
and the basis of the whole system. This they called a metre.
It is equal to about 39^ English inches, of which submultiples
and multiples being taken, the various denominations of length
are formed.
A miUimetre it the IQOOth part of a metre ,03937
A centimetre the lOOtb part of a metre ,S9371
A decimetre the 10th part of a metre 3,93710
A METRE 39,37100
A decametre 10 metres 393,71000
A hecatometre 100 metres 3937,10000
A chiliometre 1000 metres 39371,00000
A myriometre 10000 metres 393710,00000
A grade or degree of the meridian equal to
100000 metres, or 100th part of the quadrant 3937100,00000
Mb.
10
re.
2
InJlt.
9,7
109
1
1
4
213
1
10,2
6
1
156
6
134 Appendix.
The decametre is
The hecatometre
The cbiliometre
The royriometre
The grade or decimal degree of the
meridian 62 1 23 2 8
Measures of Capacity.
A cube, whose side is one tenth of a metre, that is, a cubic
decimeter, constitutes the unite of measures of capacity. It is
called the litrcj and contains 61,028 cubic inches.
iEni^. Cab» b* Dee.
A millititre or 1000th part of a liure ,06103
A centilitre 100th of a litre * ,61028
A decilitre 10th of a litre 6,10280
A litrej a cubic decimetre 61,02800
A decalitre 10 litres 610,28000
A hecatolitre lOOtl litres 6102,80000
A chiliolitre 1000«( litres 61028,00000
A myriolitre lOOOOt) litres 610280,00000
The English pint, wine measure, contains 28,875 cubic inches.
The litre therefore is 2 pints and nearly j- of a pint.
Hence,
A decalitre is equal to 2 gal. 64-3nsnr cubic inches*
A hecatolitre 26 gal. 4-3^ cubic inches.
A chiliolitre 264 gal. -^nsr cubic inches.
Weights.
The unit of weight is the gramme. It is the weight of a quan-
tity of pure water, equal to a cubic centimetre, and is «qual to
16,444 grains Troy.
Gr. Dee.
A milligramme is 1000th part of a gramme 0,0154
A centigramme 100th of a gramme 0,1544
A decigramme lOth of a gramme 1,5444
Agramme^ a cubic centimetre 16,4440
A decagramme 10 grammes 154,4402
A hecatogramme 100 grammes 1544,4023
JVeti; Frendi Weights and Measures. 135
A chiliogramme 1000 grammes 15444,0234
A myriogramme 10000 grammes 154440,2344
A gramme being equal to 15,444 grains Troy.
A decagramme 6dwt. 10,44gr. equal to 5,65 drams Avoirdupois.
lb. 02. dr.
A hecatogramme equal to 3 8,5 avoird.
A chiliogramme 2 3 5 avoird.
A myriogramme 22 1 15 avoird.
100 myriogrammes make a tun, wanting 321b. 8oz.
Land Measure.
The unit is the are^ which is a square decametre, equal to 3,95
perches. The declare is a tenth of an are, the centiare is 100th
of an are, and equal to a square metre. The milliare is 1000th
of an are. The dccare is equal to 10 ares; the heeatare to 100
ares, and equal to 2 acres, 1 rood, 35,4 perches English. The
chiliare is equal to 1000 ares, the mjriare to 10000 ares.
For fire-wood the stere is the unit of measure. It is equal to
a cubic metre, containing 35,3171 cubic feet English. The de-
cistere is the tenth of a stere.
The quadrant of the circle generally is divided like the fourth
part of the meridian, into 100 degrees, each degree into 100
minutes, and each minute into 100 seconds, &c. so that the same
number, which expresses a portion of the meridian, indicates
also its length, which is a great convenience in navigation.
The coin also is- comprehended in this system, and made to
conform to their unit of weight. The weight of the franc^ of
which one tenth is alloy, is fixed at 5 grammes ; its tenth part is
called d6cime^ its hundredth part centimcn
The divisions of time, soon after the adoption of the above,
underwent a similar alteration.
The year was made to consist of 12 months of 30 days each,
and the excess of 5 days in common and 6 in leap years was con-
sidered as belonging to no month. Each month was divided
into three parts called decades. The day was divided into 10
hours, each hours into 100 minutes, and each minute into 100
seconds. This new calendar was adopted in 1793 ; in 1805 it
136
Appendix.
was abolished, and the old calendar restored. The weights and
measures are still in use, and will probably prevail through-
out the continent of Europe. They are recommended to the
attention of every civilized country ; and their general adoption
would prove of no small importance to the scientific, as well as
the commercial world.
4t
3
2
4
li
li
10
Scripiure Long Measure.
Digit
Palm
Span
Cubit
Fathom
Ezekiel's reed
Arabian pole
Schcenus, measuring line
Eng. Feet In. Dec.
0,912
1
7
10
14
145
3,648
10,944
9,888
3,552
11,328
7,104
1,104
N. B. There was another span used in the East, equal to i
of a cubit.
Grecian Long Measure reduced to English.
4
H
n
4
100
8
Dactylus, Digit
Doron, Dochme, Palssta,
Lichas
Orthodoron
Spithame
Pous, foot
Pygme, cubit
Pygon
Pechus, cubit larger
Orgyia, pace
Stadium v^ /. «
Aulus >f"rf<»'S
Eng. paces. Feet. Id. Dec.
0,7554H
3,021 8f
7,5546|
8,3101 yV
9,0656i
1 0,0875
1 1, 59841
1 3,109|
1 6,13125
6 0,525
100 4 4,5
805 5
Milion, Mile
N. B. Two sorts of long measures were used in Greece, viz.
the Olympic and the Pythic. The former was used in Pelopon*
nesus, Attica, Sicily, and the Greek cities in Italy. The . latter
was used in Thessaly, lUyria, Phocis, and Thrace.
t These numbers show how many of each idenominatioa it takes
to make one of the next following.
Tables of Weights and Measures. 137
I'he Olympic foot, properly called the Greek foot, according to
Dr. Hutton, contains 12,108 English inches,
Folkes, 12,072
Cavallo 12,084
The Pythic foot, called also natural foot, according to
Hutton, contains 9,768
Paucton, 9,731
Hence it af pears, that the Olympic stadium is 201^ English
yards nearly ; and the Pythic or Delphic stadium, 162^ yards
nearly ; and the other measures in proportion.
The Phyleterian foot is the Pythic cubit, or 1^ Pythic foot.
The Macedonian foot was 13,92 English inches ; and the Sicilian
foot of Archimedes, 8,76 English inches.
Jewish Long or Itinerary Measure.
Eng. Miles.
Paces. Ft. Dec.
400
Cubit
1,824
6
Stadium
146 4,6
2
Sabbath day's journey
729 3,0
3
Eastern mile
1
403 1,0
8
Parasang
4
153 3,0
A day's journey
33
172 4,0
Roman Long Measures reduced to English.
£n^. Paees. Feet. In. Dec.
H
Digitus transversus
0,725|
3
• Uncia, or Inch
0,961
4
Palma minor
2,901
U
Pes, or Foot
11,604
H
Palmipes
1 2,505
1*
Cubitus
1 5,406
2
Gradus
•
2 5,01
125
Passus
4 10,02
8
Stadium
120
4 4,5
Milliare
967
N* B. The Roman measures began with 6 scrupula = 1 sicili-
cum ; 8 scrupula = 1 duellum ; 1^ duellum = 1 seminaria ; and
18 scrupula = 1 digitus. Two passus were equal to 1 decempeda
18
138
Appendix.
Attic Dry Measures reduced to English,
^
Pecks. GaU. Pints. Sol. In.
10
Cochliarion
0,276^
li
Cyathus
2^63^
4
Oxybaphon
4,144|
2
Cotylus
16,579
1*
Xestes, sextary
33,158
48
Chcenix
1 J6,705i
Medimnus
4 6 3,501
Attic Measures of Capacity for Liquids, reduced to English
Wine Measure^
Gal. Pints. Sol. In. Dec.
2
H
2
2
1*
4
2
6
12
Cochliarion
Cheme
Myston
Concha
Cyathus
Oxybathon
Cotylus
Xestes, sextary
Chous, congius
Metretes, amphora
10
tH 0,0356t%
ih 0,07 12|
-h 0,089ii
TsV 0,178H
iV 0,356«
* 0,535|
i 2,1411
1 4,283
6 25,698
2 19,626
Others
reckon 6 choi =■ 1 amphoreus, and 2 ampborei ■■ 1
keramion c
>r metretes. The keramion
is stated by Paucton to
have been
equal to 35 French pints,
or 8i English gallons, and
the other i
neasures in proportion.
Measure
9 of Capacity for Liquids ^ reduced to English Wine
Measure.
4
2
2
2
6
Ligula
Cyathus
Acetabulum
Quartarius
Hemina
Sextarius
Gal. Pints. Sol.In.Dec.
-h 0,117A
tV 0,469|
\ 0,704^
^ 1,409
1 2,818
1 5,636
4
2
50
Congius
Urna
Amphora
Culeus
7 4,942
3 4i 5,33
7 1 ie,66
143 3 11,095
Tables oj Weights and Measures.
139
Jewish Dry Measures reduced to English.
Fecks. Oal. Pints. Sol. foch.
20
Gachal
0,1 i^J- 0,031
1*
Cab
2| 0,073
3*
Gomor
5tV 1,211
3
Seah
1 1 4,036
5
Epha
3 3 12,107
2
Letteeh
16 26,600
* Chomer, cOron
32 1 18,969
wiih Measvrei of Capacity for Liquids^ reduced to Englith
Wine Measure.
•
Gal. Pints. Sol. Inch.
H J
Caph •
i 0,177
4
Log
1 0,211
3
Cab
^ 0,844
2
Hin
1 2 2,633
3
Seah
2 4 6,067
10
Bath, epha
7 4 16,2
CoroD, chomer
76 6 7,626
Ancient Roman Land Measure,
100 Square Roman feet
r= I Scrupulum of land
4 Scrupula
=: 1 Sextulus
l\ Sextulus
= 1 Actus
6 Sextiili or 5 Actus
= 1 Uncia of land
6 Udcis
= 1 Square Actus
2 Square Actus
= 1 Jugerum
2 Jugera
= 1 Heredium
100 H
eredia
=: 1 Centuria
' N. B. If we take the Roman foot at 11,6 English inches, the
Roman jugerum was 6980 English square yards, or 1 acre 37j>
perches.
Roman Dry Measures reduced to English.
4 Ligula
l^ Cyathus
4 Acetabulum
2 Hemina or Trutta
8 SexCirius
2 I Semiraodius.
Modius
5Ck.
Gal.
Pint.
Sol. In.
chV
0,01
OiV
0,04
OJ
0,06
Oi
0,24
1
0,48
1
3,84
1
7,68
4
HO
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Table of Hold and Stiver Coins.
141
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Appendix*
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ASt
INTRODUCTION
TO TBB
ELEMENTS OF ALGEBRA,
BSuavxD roR thb vii ov those
WHO ARB ACqUAINTED ONLY WITH THE FIRST PRINCIPLB8
OF
ARITHMETIC.
«m.ECTSB TKOX THB iXfiXBBA Of SVLXB*
Second Edition.
CAMBUroGE, N. ENG.
PRINTED BT BILLIARD AND METCALF,
At Om UniTenitj Frets.
SEOLV BT W« BIUUAmi>9 CilXBBIIN»E» AND BT CjniKIVBB AHO HILtTABP,
HO* 1 COBBBlLLy BOtTOH.
1821.
DISTRICT OF MASSACHUSETTSt TO WIT<
DiHrid Clerk?* Office.^
B£ IT REMEMBERED, Ttet aik tfat ninth tfay of EeHnnrr A. D. iBir, tnd in the fbity
leeond year of the Independence of the United States of Amenee, JOHN FARRAR of the
«dd diftriet ha* deposited in this office the title of n book, the right w]|eivQf he eUbni as
propriecor, in the words fbUowinf^, viz.
*< An Introdoetion to the Elements of AIgebim,.des^aied flMrtheoseof those who are aequainted
only with the first prin^ies of Arithraetie. Seleeted mm the Algebim of Enler***
In conformity to the Act of the Congress of the United States, entitled, ** An Act flnr the
cneooragement of learning, by seenring the eopic* of maps, efaarts, and books, to die authors and
proprietors of saeh copies, daring the times therein mentioned y* and alao to an act. entitled,
** An act, supplementary to an act,entitlcd. An aet fbv the encouragement of learning, by leear-
ing the eojutts of majn, charts, and books to tiie aution and proprietors of such copies during
die tiroes therein mentioned ; and extending the benefits thereof to the arts of derigning, engrav-
ing, and etching hittorieal and other prints.**
JVC. W. DATIS,
Clerk ^thc Dktriet ^ MMUKkuatU*
Ol
ADyfiRTI&BMteNT.
Now* frtit tlio^e wIk^ are ju«t eWeritf^ tipott
Ae study of Mathematics lieedf to h6 ihibi^tiied
of tlici high eharacter of Buler's Algebpfi. It
Bas* been* allowed to hold the very fifst phice
itriiong elemental^ works upon thi^ subject*
The author was* a man of geniiis. He did not, like
ftost writdrs, cotnpile from others. He wrote
firom bis own reflections. He simplMed and im-
proTed what was known, and ^dded much that
Was ne^. He is particularfy distinguished for the
dleai^iless atnd comprehensi^'eness of his views.
He seems to haVe the subject of which he treats
present to his mind in all Its relations and
bearings before he begins to write. The parts
of it are arranged in the most admirable order.
ISttch step is introduced by the preceding, and
leads to that which follows, and the whole taken
together constitutes an entire and connected
piece, like a highly wrought story.
This author is remarkable also for his illus-
trations. 0e te'aches by instances. He presents
one example after another, each^ evident By
Ti Advertisement
itself, and each throwing some new light npoa
the snhject, till the reader begins to anticipate
for himself the truth to be inculcated.
Some opinion may be formed of the adapts^
tion of this treatise to learners, from the cir«-
cumstances under which it was composed. It
was undertaken after the author became blind,
and was dictated to a young man entirely with-
out education, who by this means became - an
expert algebraist, and was able to - render the
author important services as an amanumisis*
It was written origindly in German. It has
since been translated into fiussian, French, and
Englishf with notes and additions.
The entire work consists of two volumes
octavo, and contains many things intended for
the professed mathematician, rather than the
general student It was thought that a select
tion of such parts as would form an easy intro-
duction to the science would be well received,
and tend to promote a taste for analysis among
the higher class of students, and to raise the
character of mathematical learning.
' Notwithstanding the high estimation in which
this work has been held, it is scarcely to be met
with in the country, and is very little known in
England. On the continent of l.urope this
aiithor is the constant theme of eulogy. His
writings Jiaye the^ character of classics. They
ar9 re|^arded at the samQ time as the ifiiost
V ■
,/
I
Advertisement* vil
profound and the most perspicuous, and as
affording the' finest models of analysis. They
furnish the geions of the most approved ele«
mentary works on the different branches of this
science. The constant reply of one of the first
mathematicians^ of France to those who con-
sulted him upon the best method of studying
mathematics was, '^ study Euler.^^ ^' It is need-
less,'^ said he, ^ to accumulate books ; true
lovers of mathematics will always read Euler;
because in his writings every thing is clear,
distinct, and correct ; because they swarm with
excellent examples; and because it is always
necessary to have recourse to the fountain
head."
The selections here offered are from the first
EngUsh edition. A few errors have been cor-
rected and a few alterations made in the
phraseology. In the original no questions were
left to be performed by the learner. A collec-
tion was Tuade by the English translator and
subjoined at the end with references to the
sections to which they relate. These have been
mostly retained, and some new ones have been
added.
Although this work is intended particularly
for the algebraical student, it will be found to
contain a clear and full explanation of the fun-
damental principles of arithmetic ; vulgar frac-
* Lagrange,
Till
44neru$finmf'
tions, the doctrine of roots find powers, of tbo
different kinds of proportion and progreBsion|
are treated in a mannjer, that can hardly fail t9
interest the learner a|id^ make him acquainted
with the reason of those rules which he has SQ
frequent occasion to apply.
A more extended work on Algehra formed
after the same model is new in the press and wiU
spon he pu^blished. Thb will he followed h^
other treatises upon the different hranqhes of
pure mathema|icsp
JOHN FARRAR,
P>oft»or •€ ICatbematki and Nmtaml nSktufkf in tkt
eambiidge, February, 181S.
CONTENTS.
gECnOKL
idr ras mrvBRBHT icxinoDs of oAtouLAnxa sncpui <|VAimEmt».
CHAPTER L
(y^iMiiUki^ma^tnifefifraL t
CHAPTER IL
BxplaMUm cf the SKgns + Plus and -— Minns. S^
CHAPTER IH.
Of the MvUiiOUiQHan tf 5lmpte quanfiHes. 7
CHAPTER IV.
Qf the nalure of whole Jinmben or Integers, with retpect to
their Factors. 1 1
CHAPTER V.
Of the Division of Simple (fuafUitiBS. 13
CHAPTER VL
Cjf the properties cf Integers with respect to thdr Divisors. 16
CHAPTER VII.
Of Fractions in generaL 19
CHAPTER Vm.
(^ the properties of Fractions. 24
. CHAPTER IX.
Of the JiddiHon and Subtraction of FfOdions. S5
• CHAPTER X.
Of the MuUipUcation and Division of Fractions. 29
CHAPTER XI.
Of Square JV^mfters. 3S
CHAPTER XII.
Of Square BootSf and of Irraiional'M^ifmbers resuUingJrom
them. S5
CHAPTER XIII.
Of ImpoBBitU or Imaginary Quantitiei, which ari§efrom the
$ame source. 40
CHAPTER XIV.
Of CvMc JV\»m^s. 45
CHAPTER XV.
Of Oube RooU, and of Irrational Mnniers resisting fro^ 44
CHAPTER XVI.
Qf Powers in general. 47
CHAPTER XVIL
<)f the cakulaHon of Powers. 51
CHAPTEfl XVIIL
Of Boots with rdation to Powers in genoraL 59
CHAPTER XIX.
Of the Method cf representing Irraiiawl ^tSumbers by FraeHonal
ExponetUs. 6§
CHAPTER XX.
Of the different Methods of Calculation, and of their mutual
^ Connexion. St
. SCCTIOir SBCOSTD.
OV THE DIFFEnENT METHODS OF OALOULATIKG OOMPOUHD QUAMTITIBS.
CHAPTfeR I.
Of the JddUion of Compound ^lantities. 61
CHAPTER 11.
Of ihe Subtraction tf Compound Quantities. 6$
CHAPTER 111.
Of the MultipUcation of Compound (luaraities. 64
CHAPTER ir.
Of the Bvoision of Compound Quantities. « 70
CHAPTER V.
Of the Besolution tf Fractions into injimte Berks. 74
CHAPTER VI.
Of the Squares of Compound Quantities. 83
CHAPTER Vn.
Of the Extraction of Boots applied to Compound ((uatttUies. 86
CHAPTER VIII.
CHAPTER IX.
Of Cubes, andofihe Extraetim of Cube Moats. 94
CHAPTER X.
Of the higher Powers of Compound ^uanHHes. 96
CHAPTER XL
Of the Transposition of the LeUerSf on which the demonstra*
tUm rf the preceding BuU is founded. 10ft
CHAPTER XIL
Of the expression of Irrational Powers by Infinite Series. 106
CHAPTER Xni.
Of the resotution if Jftgatvoe Powers. 109
SfiCTIOM THIRD.
«
OF RATIOS AND PROPORTIONS.
CHAPTER I.
Of Arithmetical BaHo, or of the difference between two Jnim-
bers. 115
CHAPTER IL
Of JMihmdical Proportion. 115
CHAPTER in.
Cf Jhrithmetical Progressions. lit
CHAPTER IV.
(ff the Summation of Jirithmetical Progressions. 12S
CHAPTER V.
Of Oeomelrieal Batio. 1&6
CHAPTER VL
Of the greatest Common Divisor of two given numbers. 1S8
CHAPTER VII.
Of Oeometrical Proportions. 132
CHAPTER VIII.
Observations on the Btdes of Proportion and their utUUy. 135
CHAPTER IX.
Of Compound BdaUons. 140
Ill (ktOenU.
CHAPTER!.
Qf QtomOrieal Pn^gftsshtm. 146
CHAPT£B XI.
Of If^niie Decimal FracHans. 15ii
8BCTION FOUBTH.
•r ALOEBRAIO BqUATIOKS, AND OF THB MCSOLIHION OF TH08B
B<4UATiOV8.
CHAPTER h
(y(heB(iiabnafPr€blm$ingemraL 158
CHAPTER n.
Of the BetobMon if SimfU EjikUimiif ar JBguattoiu ^ ike
fini i^ree. 161
CHAPTER HI.
OftheB(iiiitumrf^e$lwmrdMnfio^ 165
CHAPTER IV.
OfiheBesolMhimqfiwoarmm^Eqimtumsift^ 175
CHAPTER V.
4)fiheBei(iiiiUmqfFureifiiuidm^ 184
CHAPTER YL
iQf the Rt$ob»tum if Mixed Bqtiatians of the Second Dqpree^ 190
CHAPTER VIL
iff ihe MOure ef EquaUanB of (he Second Degree, 198
ifnestumsfor Practice. d04
Jflnv demonstroHona of the fundamental proposUione of the
&thbook(f EudUdU Elements. %IT
i
INTRODUCTION
TO TBI
ELEMENTS OF ALGEBRA.
SECTION L
or THE DIPFKUBMT METHODS OF OALCULATIOK APPLIED TO SIMPLE
QUANTITIES.
CHAPTER I.
Of MaihemaHei in gtturaL
Whatetbr is capable of increaso or dimioution/ is called
magnittide ur quantityl
A sum of moneyy for instance^ is a quantity^ since we may in-
crease it or diminish it The same may be said with respect to
any given weighty and other things of this nature.
£• From this definition^ it is evident^ that there must be so ma-
ny different kinds of magnitude as to render it difficult even to
(numex^te them : and tiiis is the origin of the different branches
of mathematics, each being employed on a particular kind -
of magnitude. Mathematics, in general, is the science of qwm^
tUy ; or the science which investigates the means of measuring
quantity.
S. Now we cannot measure or determine any quantity,
except by Considering some other quantity of the same kind
as known, and pointing out their mutual relation. If it were
proposed, for example, to determine the quantity of a sum of
money, we should take some known piece of money (as a dollarf
a crown^ a ducat, or some other coin,) and shew bow many of
G
2 MgOrk. Sect 1*
these pieces are contained in the giiren sum. In the same mnn^
ner, if it were proposed to determine the quantity of a weighty
we should take a certahi known weight ; foi* example^ a pound*
an ouncCf &c«9 and then shew how many times one of these
weights is contained in that which we are endeayoaring to as-
certain. If we wished to measure any length or extension^ we
should make use of some known length, as a font for ezamplet.
4. So that the determination, or the measure of magnitude of
all kinds, is reduced to this : fix at^ileasure upon any one known
magniiude of the same species with that which » to be deter«
mined, and consider it as the measure or unit ; then, determine
the proportion of tlie proposed magnitude to this known mea*
sure. This proportion is always expressed by numbers ; sor
that a number is nothing but <^e; proportion of one magnitude
to another arbitrarily assumed as the unit.
5. From this it appears, that all teagnitudes may be expressed
by numbers ; a^id that the foundation of all the matbematical
sdences must be laid in a oomfilete treatise on the science off
numbers ; and in an accurate ^camitialion of the different pos«
aible methiids of cakuiation.
This fundamental part of mathematics is called analysis, or
fdgebnu
6. In algebra then we consider only numbers which repre«
sent quantities, without regarding the diftrent kinds ef quantity.
These are the subjects of other branches of the mathematics.
7. Arithmetic treats of numbers in particular, and is the
sdtnce vf fiumber$ pr^perlfi so eaUed ; bet this science extends
only to certain methods of calculation whi«^h occur in rommoe
practice: algebra, on the contrary, comprehends in general
all the cases which can exist in the docti-ine and calculation of
numbers*
Ghap.8. Of Bimth ^wmttHes^ 9
CHAPTER IL
<
Erplanafum oj the signs + plus and — minas.
%, Whbh we have to add one given namber to another* this
18 if^dacated by the sign + which is placed before the second
number, and is read plus. Thus 5 + S signifies that we must
add S to the nuoiber 5, and every one iLnows that the result is
a ; in the same manner 12 + 7 make 19 ; 25 + 16 make 41 ; the
anm of 25 + 41 is 66f to.
9. We also make use of the same sign + or j^/us, to connect
seTeral numbers tigetker ; for example* T +5 + 9 signifies that
In the number 7 we must add 5 and also 9* which make 21. The
reader will therefore understand what is meant by
8 + 5 + IS + U + 1 + 5 + 10 ;
tias. the sum of all these numbera, wh irk is 51.
10. AU this is evident ; and we have only to mention, that*
HI algebra* in order to generalise numbers, we represent them
by letters* as a* A, e^ d. Ice. Thus the expression a + b signifies
the sum of two numbers, which we express by a and 6, and
tbese nuiibers may be either veqr great or very small, lii the
same manner*/ + m+b + Xf signifies the sum of the numbers
represented by these four letters.
If we know therefore the numbers that are represented by
letters* we shall at all tiflies be able to find by arithmetic* the
sum or value of similar expressions.
11. When it is requiiped, on the contrary* to subtract one
given number from another, this operation is denoted by the
aign -*-* which signifies wiwuSf and is placed before the number
to be subtracted : thus J^— 5 signifies that the number 5 is to bo
taken from the number 8 ; which being done, there i-emains 3«
In like manner 12 — 7 is the same as 5 ; and 20 — 14 is the same
as 6, &c.
1 2. Sometimes also we may have several numbers to be subtract-*
ed from a single one ; as for instance, 50 — 1 — d^-5 — 7 — 9.
This signifies, first* take 1 from 50, there remains 49 ; take 3 from
that remainder, there will remain 46 ; take away 5* 4 1 remains ;
take away T* 34 remains ; lastly* from that take 9^ and there
4 Mgeira. Seet 1.
# *
remains 25 ; this last remainder is tbe value of the expression* *
But as the numbers If 3, 5, 7, 9, are M to be subtractedf it is
the same thing if we subtract their sum, which is 85^ at once
from 50, and the remainder wiD be 25 as before.
13. It is also very easy to determine the valtie of similar
expressions, in which botli the signs + P^^ ^nd — mnu$ are^
found : for example ;
12 — 3 — 5 + ^^^lhthe same as 5.
We have only to collect separately the sum of the numbers that
. have tlie sign + before them, and subtract from it the snoi of
those that have the sign — • The sum of 12 and 2 is 14; that
of 3, 5 and 1, is 9 ; now 9 being taken from 14, there remains 5;
14. It will be perceived from these examples tliat tAe order
in ivhich we write the numbers is quite i/ndifferent and arbUrmy,
provided the proper sign -^ each be preserved. We might with
equal propriety have arfthged the expression in the preceding
article thus ; 12 +2 — 5 — S— 1, or 2 — 1 --3 — 5 + 12, or 2+
12 — 3 — 1 — 5, or in still different orders. It must be observed,
that in the expression propdsed, the sign + is supposed to be
placed before the number 12.
15. It will not be attended with any more dtScuify, if, in
order to generalize these operations, we mnike use of letters
instead of real numbers. It is evident, for example, that
a — b — c + d — e
signifies that we have numbers expressed by a and d, and that
from these numbers, or from their sum, we must subtract the
numbers expressed by the lettei*s 6, c, e,' which have before them
the sign — •
16. Hence it is absolutely necessary to consider what sign is
prefixed to each number : for in algebra, simple quantities . ore
numbers considered with regard to the signs which precede, or
affect them. Further, we call those positive quantities, before
which the sign -f is found ; and those are called negative quan-^
iities, which are affected with the sign — •
17. The manner in which we generally calculate a pcrson^s
property, is a good Illustration of what has just been said. We
denote what a man really possesses by positive numbers, using,
or understaning the pign + j whereas his debts are represent-
/"
L
dl by DOgatiTe nttiDbera» or by usios the dgn -^^ TkuSf when
it is said of any one tbat be baa 100 crowns, but owes 50, this
iseaas that his property really amoonts to 100>^ 50 ; or, which
is the same thing* + lOO -* 50* that is t» say 50.
18. As negative nambers may be considered as debtSf because
Msitive pumbers represent real possessions, we may say that
negative numbers are less than nothing. Thust when a man
has nothing in the world, and even owes 50 crowns, it is certain
that he has 50 crowns less than nothing ; for if any one were to
make him a present of 50 crowns to pay his debts, he would
still be only at the point nothing, though really richer than
before.
19. In the same manner therefore as positive numbers are
incontestably greater than nothing, negative numbers are less
ttan nothing.* Now we obtain positive numbers by adding 1 to
0, that iet to say, to nothing; and by continuing always to in*
crease thus from unity. This is the origin of the series of num-
bers called natural numbers ; the following are the leading terms
of this series :
0, + !, + «, + S, + 4,+ 5, + 6, + 7, + 8, +9, + 10,
and so on to infinity.
But if instead of continuing this series by successive additions,
we continued it in the opposite direction, by perpetaally sub*
tracting unity, we should ha^e the series of negative numbers :
0, — I, — 2, — 3, —4, — 5,— 6, — r, — 8, — 9, — 10,
and so on to infinity.
* By bei&p lest than nothing is meant simply thtt they are of such a nature
as to cancel or destroy an equal number with the sign plus before it, so that
-—4^ or— a is as really a positive thing, and is as easily conceived, as +4 or
+a» The quantity 4 or a may be considered independently of its sign. The
sign 4- implies that this quantity is to be added, and the sign — Out it is to
be subtracted. This subject may be illustrated by the scale of a thermome-
ter. After observing the mercury to stand at 50°, for instance, I am told,
that it has changed 4^, I have a distinct idea of the portion of the scale de-
noted by four of its divisions, without applying tliem in any particular direc-
tion. Hut when I am further informed that this change of the thermometer
is -" or tubtractive with respect to its former state, 1 then unilcrstitnd that
the mercury stands at 46^, Avbereas it would be at 54^ if tlie cliangc had been
+ or additive.
SO. All these numbers, whether positire or nej^tive, liave the
knnwti appellatioD of whole numbers, oi* integemf which conse-
quently are either greater or less than nothing. We call tliem
ivtegerg^ to distiifguisb them from fractions, and from several
other kinds of numbers of whtoh we shall hereafter spei^. For
histanoe, 50 being greater by an entire unit than 49t it is easjjr
to comprehend that there may be between 49 and 50 an infinity
of intermediate numbei-s, all greater than 49, and yet all lesfl
than 50. We need only imagine two lines, one 50 feet the
other 4d feet long, and it is evident Ihat there may bedi*awn an
ininlte number of lines all kinger than 49 feet, and yet shorter
than 50.
£1. It is of the utmost importance through the whole of
dgebra, that a precise idea be funned of Aose negative quanti-
ties about which we have been speaking, i shall content myself
^ith remarking^ here that all audi espressionB^ an
+ I— l,+2— 2, + S — S, + 4— 4,&C.
are equal to or nothing. And that \
.f 2 — - 5 is eqoal<to -— 8*
,Fer if a person has S crowns, and owes 5, he has not only
nothing, but still owes 5 crowns : in the same manner,
7 — 12 is equal to— - 5, and 25 — 40 is equal to — ^ 15.
22. The same observations bold true, when, to make the
expression more general, letters are used instead i^ nufnbers :
or nothing will always be the value of + a — a. If* we wish to
know the value -fa — h two cases are to be considered.
The first ie when a is greater than h; h must then be sub-
tracted from a, and the remainder (before which is placed or
understood to be placed the sign +) shews the value sought.
The second case is that in which a is less than i : here a is
to be subtrarted from 5, and the remainder being made negative^
by placing before it the sign — p will he the value sought.
^Vj
Cbftp.i.1 Of SHmfk ^mitUia. %
CHAPTER lit.
Of the MuUipUcatim of Simple Quantities*
£3. WhBw there are two or more equal numbers to be added
together, the cxpreasiou of their sum may be abridged i for
maniple,
a+ah the same with 2 x Of
a + a+a 3x Of
a^a + a+a 4 x <if and so on $ where x 10 the sign
of multipiication. In this manner we may form aa idea of mal«
tiplication ; and it is to be observed that^
' d X ci signifies 2 times, or twice m
dxa Stimesy'or thrice a
4 X a 4 times a, &c.
24. J^ therefore a mmber exffened fty a letter is to he wmUiplied
hy any other manber, we simply put thai mtiakr before the leUeri
thus,
a multiplied by SO b expressed by 20 a, and
6 multiplied by SO givea 30 1, &c«
It is evident ahio that taken onrOf or 1 c, is just c
25. FVipthf r, it is extremely easy ti> raidtiply such producfl
again by other numbers -, for example :
2 times, or twice 9 a makes 6 a^
3 times, or thrice 4 b makes 12 ft^
5 times 7 x makes 35 x,
and these products may be still multiplied by other numbers at
pleasure.
26. frhen the nnntfter, by ivhidi we are to mtuHptyf is also re*
presented by a lettert we place it immediately before the other letter ;
thus, in multifdying b by a, the product is written oft ; and p q
will be the product of the multiplication of the tiumber q by p.
If we multiply this p q again by a, we shall obtain apq»
* 27. It may be remarked here, that the order in which the letters
arejdned ios;ether is indifferent ; that n 6 is the same thing as 6 a ;
for h multiplied by a produces as much as a multiplied by ft.
To understand this, we have only to substitute for a and b
8 JUgAra. Soctl.
known numbers^ as 3 and 4 ; and fhe truth will be selfieTident |
for 3 times 4 is the same as 4 times 3,
28. It will not be difficult to perceive^ that when you have to
put numbers, in the place of letters joined together, as we have
described, they cannot be written in the same manner by put-
ting them one after the other* For if we were : to write 34 for
3 times 4, we should have 34 and not 1£. When therefore it is
required to multiply common numbers, we must separate them
by the sign Xy or points : thus, 3 x 4, or 3*4, signifies 3 times 4,
that is 112. So, 1 X S is equal to 2 ; and 1x^X3 makes 6. In
like manner 1x2x3x4x56 makes 1344 ; and 1x2x3
X4x5x6x7x8x9xlOis equal to 3626800, &c.
29. In the same manner, we may discover the value of an
expression of tdis form, 5 " 7 • 8 a t c d. It shews that 5 must be
multiplied by 7, and that this product is to be again multiplied
by a $ that we are then to multiply this pnoduct of the three
immbers by a, next by &, and then by c,aiid lastly by d. It may
be obsariced also, that instead of 5 x 7 x 8 we may write its value^
280 ; for we obtain this number when we multiply the product
of 5 by 7 or 35, by 8.
30. The results which arise from the multiplication of two or
laore numbers are called proiujcU ; and the numbers^ or indivi-
dual letters, are called /actors.
31. Hitherto we have considered only positive numbers, and
there can be no doubt, but that the pr^ucts which we have-
seen arise are positive also : viz. +ahj + h must necessarily
give +ah» But we must separately examine what the multipli-
cation of + a by *-« &> and of — a by — (, will produce.
32» Let us b^n by multiplying i— a by 3 «r + 3 ; now since
— a may be considered as a debt, it is evident that if we take
that debt three times^ it most thus become three times greater,
and consequently the required product is — 3 a. So if ^e multi-
ply — a by -f 5, we shall obtain — io, or, which is the same thing,
—at. Hence we conclude, tiiat if. a positive quantity he multi-
plied by a negative quantity, the product will be negative;
and lay it down as a rule, that + by -f makes +, or jilta, and
that on the contrary +by •— i or — by + gives •— , or minus.
^
Chap. S. Of Bimpk ^iAntUies.
33. It remains to resolve the case in wliich — • is multiplied by
— : or^ for ^xample^ — a by — b. It is evident, at fii^st sight,
vrith regard to the letters, that the product will be a 6 ; but it is
doubtful whether the sign +, or the sign — » is to be placed before,
the pn duct ; all we know is, that it must bo one or the other of
these signs. Now I say that it cannot be tlie sign — : for — a
by + fr gives -—aft, and — a by — 6 cannot produce the same re-
sult as—- a by + 6; but must produce a contrary result, that is to'
aay, +ah; consequently we have the following rule : — « multipli-
ed by •— produces +, in the same manner as -f multiplied by +.*
* It is a subject of g^reat embams&ment and perplexity to learnen to cor-
eeiTC how the product of two negative quantities should be positive. This
arises from the idea they receive of the nature of multiplication as explained
and applied in arithmetic, where positive quantities only are employed. I'he
term b used in a more enlarged sense when negative quantities are ooncemed,
as may be shown without making use of letters. If 1 wished to multiply, for
instance, 9 — 5 (or 9 diminished by 5) by 3, 1 should first find tlie product of
9 by 3 or 27. But tliis is evidently taking the multiplicand too great by S,
and of course the product too great by 3 tiroes 5 ; I accordingly write for the
product 27— 15, equivalent to 13, which is the product that would arise
from first performing the subtraction indicated by the sign — , and using the
result as the multiplicand* Thus,
Multiplicand 9—- 5 which is equal to- 4
Multiplier 3 3
Product 27—15 which is equal to 12
Let us now take for the multiplier the quantity 7<— 4^ which is equivident
to 3. We multiply, in the first place, by 7, in the manner Uiat we have
just done by 3, and the result is 63 — 35. But as the multiplier is 7 diminish*
cd by 4, multiplying by 7. must give 4 times too much. Accordingly we take
4 times the multiplicand, or 36 — 20, and subtract this from 63 — 35, or 7
times the multiplicand. Now in making this subtraction it is to be observed
that the subtrahend 36 — 20 is 36 diminished by 20, and if we subtract 36 we
take away too much by 20^ and must therefore add this latter quantity. Con*
aequently the true product will be 63 •— 35 — 36 + 20, equivalent to 12, as be
fore. Thus tliis mode of proceeding gives the same result as that obtained
by first performing the subtractions indicated in the latter term of the multi-
pltctnd and multiplier. The several steps in eadi case-are as follows :
Multiplicand 9 — 5 which is equal to 4
Multiplier 7 — 4 which is equal to 3 «>
63—35 Product 12
— 36 + 20
I'roduct 63 — 35 — 36 + 20 or 83 — 71^ that is 12.
Ml 4lg* 2
1^ Sffl^a. 860. t,
34. The ipfiles wfaiich we kuve explained aro ezpresped more
briefly as follows ; ' ,
Thus we ftee that 7 or +7 by— 5 givei «— 35» and — 4 by +9 giFCS — 3^,
and — 4 by — 5 g-iTes + 20. The same general reaioning wUl apply when let.
ters are used ihstead of numbers. '
Maltiplioand a-^h
MiillipUer ^ C'^ d
Product ac — bc^-ad+bd.
We aay in this case that, vhen we multiply a by c we tajke the multiplieand
too great by 6, and must therefore diminish the reaalt a c by the product of 6
l^y « or 6 c* So also in multiplying the t^ro teni^ of the multiplicand by c,
ve luLve tjUc^ the^ multiplier too great by d^ and must therefore diminish the
result a c — frf by the product of a —- 5 by <^ or a J— b </• But if we sub-
tract the whole of a d, we subtract too much by bdf bd must necordingly ,be
added.
The rule for negative quantiti^ hcve iUoslraliied is not oecessai^ where
mere numbers are employed, because the subtraction indicated may i^ways be
performed. But this cannot bf done with ijesp^ct to letters which stand for
no particular values, but are intended as general expressions of quantities.
The truth of the rule may be shown also when applied to quantities taken
singly. We aay that multiplying one quantity by another is taking one as
many times as there are units in the other, and the result is the same, which-
ever of the quantities be taken for the muUiplic^ad. Thus multiplying 9 by 3
is taking 9 three times, or, which is the same thing, taking 3 nine times
(Arithi^r)* But in aridimetic, quantities* are alway» taken affirmative^,
that is additively. When therefore we take 9 or + 9 three timey additively, or
4. 3 nine times additively, the result will evidently be additive or '^-27, When
on the odprary ooe of the ftctors is negative, as for inataoee, in multiplying
<— 5 by -(- 3 ; in this ease, — 5 is to be taken three times additively, and — > 5
added to — - S add^d to — 5 is clearly — 15. So also if we consider + 3 aa the
multiplicand, tben + 3 is to be taken five times aubtractively 1 now 3 taken
subtractively once (or which is the same thing 3 x -^ 1) 19 equivnknt io —3,
taken aubtractively twice is -^ 6, three times ia «— 9| five times is -^ 15. But,
vhen the multiplicand and multiplier are both n^^tive, as in the case of moU
tiplying — 5 by — - 4 ; here a subtractive quantity is to be taken subtractively,
that is. we are to take away successively a diminishing or lessening quantity,
which is certainly eqnivalent to adding an increasing quantity* Thus if we
take uway — 5 once, we augment the sum with which it is to be connected
by -h 5 ; if we take away — 5 twice, we make the augmentatioD 4- 10 ; if four
times, 4- 20 1 that is, «- 5 X — 4 is equivalent to + 20,
€h%)f. 4. Of BhnfU ^AnftltM. 1 1
1^— • Thus, when It is required to rioltipljr the fotlowlng
aambera i 4. Oi — ft, — c, + d ; we have flret + multiplied by
'^bf which tnakes^— ct 6 ; this by —0^ gives -f a ft «; and this bjr
4-<l»gives-Hofted«
sa. The difficulties with respect to the signs being removed^
we have only to show how to multiply numbers that are them-
* selves products* If we werOf for instance, to multiply the
humber a ft by the number e d^ (he product would be a ftc d, and it
is obtained by multiplying first oft by c, and then the result of
that multiplication by d« Or, if we had to multiply S6 by Id |
since 18 is equal to d times 4, we should only multiply 36 first by
S, and then the product 108 by 4, in order to have the whole pro-
duct of the multiplication of 18 by S6, which is consequently 4 S8«
56. But if we wished to multiply 9 oft by ded, we might write
iedySabf however, as in the present instance the order of the
numbers to be muUi(dled is indlilbrent, it will be better, as is
also the cufltom» to place the oommon numbers before the letters,
And to express the product thus : 5 x 9 a fto d, or 15 oft cd ; since S
times 8 is 15.
So if we had to multiply 12p jr by 7 ory, we should obtain
I
CHAPTER IV.
Of tlie nature qf whole number$ or integers, wWi rei^t to their
87. Wa have observed that a product is generated by the
multiplication of two or more numbers together, and that these
numbers are called JHctors. Thus the numbers o, ft, c, d, are
the factors of the product aft c d«
SB. Ifi there foret we consider all whole numbers as products
of two or moi*o numbers multiplied togethert wo shall soon And
that some cannot result from such a multiplicatluni and conse-
quently have not any factors | while others may bo tlio products
of two or more multiplied togcthorf And may consequently have
two or more factors. ThiiN, 4 is produced by 2x8; 6 by 8x8;
8 by 8x8x8; or 27 by 3x8x3; and 10 by 2x5, &c.
10 ^^ffu^ fleet. 1«
- d9^ Bttt^ imthe other ha^» the nantbersyfit 3, 5^ 7f tl, tSy
ITftxeh cftnnotlie represented int the same manner bj factom^
nnlesa for that purpose we make ase of anity^ and reiiresent 2^
for instance, b j 1 x S. Now the numbers wbieh are multiplied
by If remaining the same, it is not proper to reckon unity as a
factor.
All numbers, thi&reforfey such as S, 3, $f 7, \U 13, 17, &e*
winch cannot be represented by fiictors, are called simpUf or
prime numbers; whereas others, as 4,6, 8, 9, 10, 12, 14, 15, 16^
18, &c. which may be represented by factors, are called comfownd
numbers.
40. Shmple or prime tmrnhers deserve therefore particular
attention, since they do not result from the multiplication of two
or more numbers* It is particularly worthy of observation that
if we write these numbers in succession as they follow each other
t^^s; -• . ,
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 4f, &c.
wO'Can trace no regular order; their increments are sometimes
greater, sometimes less ; and hitherto no one has been able to
discover whether they follow any certain law or noL
41. .^I compound numberSf which may be represented byfadors,
result from the prime numbers 4tboroe mmtimed $ that is to saf/f aU
their fadars are prime nun^bers. For, if we find a fieu^tor which
is not a prime number, it may always be decomposed and r^re-
sented by two or more priro^^numbers. When we have repre-
sented, for instance, 4}ie number 30 by 5 x 6, it is evident that 6
not being a prime number, but b^ing produced by 2x3, we
]p»ight have represented ^0 by 5 x 2 x 3, or by 2 x 3 x 5 ; that
is to say, by factors, which are a|I prime numbers.
42. If we now consider those compound numbers whidi/mi^
be resolved Into prime numbers, we^hall observe a greatdiflfer-
ence among them ; we shall find that somo have only two factors,
that others have three, and others a still greater number* We
have already seen^ for example, that
4 is the same as 2x2f
8 2X2X2,
10 2x5,
14 2X7,
16 . ' •* 2X2X2X2,
6 is the same as 2x3,
9 3x3,
12 2x3x2,
15 3+5,
and so on»
€hap. 5. Cfam^ qprnKlit^. la
, 4S. Htince it ifl easy to find a metfiod for analysMig aay wm^
berf or resolving it into its simple factors. Let there be piOi
posed, for instancef tbe number 360 ; we shall represent it first
by 2 X 180. Now 180 is equal to 2 x 90, and
90^ f «X45,.
45 I is the same as «< 3x15^ and lastly
15 J t3X S.
So that the number S60 may be represented by these simple
factors, 2x2x2x3x3x5; since all these numbers multiplied
together produce 360.
44. This shews, that the prime numbers cannot be divided by
other numbers, and on tbe other hand, that the simple JadorM of
annpound numbers are foundt most conveniently and with tke great-
est ceriaintyi by seeking the simple, or prime numbers^ by whi(A
those compound numbers are divisible* But for this, diroision is
necessary ; we sholl therefore explain the rules of that operation
in the following chapter.
OHAPTEB V*
Of tbe JHvision f^ Simple QuatMies.
45. WtiEir a number is to be separated into two, three, or
more equal partis, it is ddne by means of drtision, which enables
us to determine the magnitude of one of those parts. When we
wish, for example, to separate the number 12 into tbree equal
parts, we find by division that each of those parts is equal to 4.
The following terms are made use of in this operations The
number, which is to be decompounded or divided, is called the
diddend ,* the number of equal parts sought is called the divisor i
the magnitude of one of tfatee parts, determined by the division^
is called the quotient ; thus, in the above example ;
12 is the dividend,
3 is the divisor, and
4 is the quoti^iL
46. It follows from this, that if we divide a number by 2, or
into two equal parts, one of those parts, or .^,he quotient, taken
twice, makes exactly the number proposed ; and, in the same
f^
14 Jlgdira. debt 1.
nainiiery if we hnvh tL niMber to be divided by 3, the quotient
taUn thrice ihh^ give fbe sftme numbef again. In general, ike
mulHplieation if the qubtUnt by the drdsor must always reproduce
the dividend.
47* It is for this reascm that divisioi^ is called a rule,
which teaches us td find a tiambei' or quotient, which, being
multiplied by the divisor, witl exactly produce the dividend*
For example, if &S is to be divided by 5, we seek a number
which, multiplied by 5, will produce 35. Now this number is.
7, since 5 times 7 is 35. The mode 4)f expression, employed
in this reasoning, is ; $ in 35, 7 times ; and 5 times 7 makes
35.
48. The dividend therefore may tie considered as a product^
of which one of the factors is the divisor, and the other the
quotient. Thus, supposing we have 63 to divide by 7, ^we en-
deavour to find sucli a product, that taking 7 for one of its
factors, the other factor multiplied by this may exactly give 63.
Now 7 X 9 is such a product, and consequently 9 is the quotient
obtained when we divide 63 by 7.
49. In general, if we have to divfcle a number a & by a, it is
evident that the quotient will be b ; for a multiplied by b gives
the dividend a dr. It is Clear also, that if we had to divide abhf
b, the qtiotient woiild be a. And in all examples of division
tbut can be proposed, if we divide (lie dividend by flie qvotien^
we abail again oMmi (ktf divisor; tar wM divided by 4 gives
»^ 89 94 divided by 6 wfH give 4. i
50. As tA^ whd$ opefaHen consi^ in represenHng the dividend
ky ttoo factors, of iidhiih one shaU be eqital to the dvoisor, the othe^^
iotheqaatieni ; the Mlowinj; ex«iiipl« wiH be easily andcmtoed.
1 sfty Arst, that the dividend a be, divided by a, gives ft c f for a,
multiplied by I' e, ftftivtM^ abc: in Ih^ same mftmier a be, being
divided by ft, we shall* have ae; and abc, divided by ac, gives k
I say idso, that 12 m n, divided by S m, gives 4 n ,- for 3 m, multi-
plied by 4 ft, makes limn* Bttt if this same number 12 m n had
been divided by IS, we shoiM have tibtaified the quotient m n;
51. Since every ndmber a may be expftfssed by 1 a or <me a, it
]» evideirt that if wo had ta divide a or 1 a by 1, the quotient wouM
Cbap.{». Of Bimpb quoMtiiks. is
lie tde same number a. Bnt» oq tbejcontraiyt if the aame aamber
Q,or la is to be divided b^ a, the quotient will be 1.
52. It often happens that we cannot represent the dividend as
the product of two factors^ of wluch one is equal to the divisor i
and then the division cannot be^ perforog^ in the manner we
ha%'e described.
When we have^ £9r e^wpofitf d4 to be divided by 7f it is at
first 8i|;ht obvious^ that the number 7 is not a factor of ^ | for
the product of 7 X S is only 21^ and consequently too Bmall, and
7 X 4 makes 28f which is greater than £4. We discover however
from thi^, that the quotient must be greater than S, and less than
4. In order therefore to determine itexactly, we employ another
species of numberSf which are called JrodwiSf and which we
ahall consider in one of the following cliapters.
53. Until the use of fractions is considered, it is usual to rest
satisfied with the whole number which approaches nearest to
the true quotient, but at the same time paying attention to the
remainder whicb is left ; thus we say, 7 in 24, 3 times, and the
reramnder is 3, because 3 times 7 produces only 21, which is 3
leps than S4« We\ may consider the following examples in the
i^ame manner :
6)34(5| that is to say, the divisor Ls 6» the dividend 34^.
30 the quotient 5^ and the remainder 4.
4
9)41(4, here the divisor is 9, the dividend 41, the quo-
36 tient 4, and the remainder 5.
5
The foUowing rule ia to be observed in examples where there
IS a remainder.
54. If lo^ wiUiflijt ihc dim^
odd ih^ remainder, vfc muet o^toin the dividend; this is the
method of proving divisibnf and of discovering whether the
cfdculatlon is right or not. Thus, in the first of the two last ,
exanqiles, if we mnttiply 6 by 5, and to the product 30 add tlici
remainder 4, we obtain 34fl or the dividend. And in the last
eiiaiiH^e» if we multiply the divisor 9 by the quotient 4, imd to
the product 36 add the remainder 5^ we obtain the dividend 41..
15 MgAra. Sect U
55. Lastl)^, it is tiecess^ to remark here, with l*ogard to thd
signs -f pita and — ininuSf that if we divide + ah by + a, the
quotient will be -f 6, which is evident But if we divide + ab
by — iif the quotient will be — 5 ; because — ax — h gives -fat.
if the divided is — a &, and is to be divided by the divisor -f a,
the quotient will be •— 6 ; because it is •— ft, which, multiplied
by +ih makes —-aft. Lastly, if we have to divide the dividend
— cr ft by the divisor — a, the quotient Will be -f ft ; for the divi^
dendfx^ a 6 ts the product of — > a by + ft.
56. WiXh regard therefore to the signs + and — , division admits
the same rules that we have seen applied in midtiptication ; viz.
+ by + requires +; + by — requires — ;
— by + requires — j — by — requires + :
or in a few words, like signs give plus, unlike signs give minus.
57. Thus, when we divide 1 8 p 9 by •— S p, the quotient is — ^6 9.
Further 5 •
— 30 a: y, divided by + 6 y. gives — 5 a?, and
—54 a ft c, divided by — 9 ft, gives + 6acj
for in this last example, — 9 ft, multiplied by -f- 6 a c, makes — 6x
9 a ft c, or -— 54 a ft c But we have said enough on the division of
simple quantities ; we shall therefore hasten to the explanation
of fractions, after having added some farttier I'emarks on tha
nature of numbers, with respect to their divisors.
CHAPTER VI.
Of the properties of integers ivUh respect to their divisors.
s
58. As we have seen that some numbers are divisible by cer-
tain divisors, while others are not ; in order that we may
obtain a more particular knowledge of numbers, this difference
itiust be carefully observed, both by distinguishing the numbers
that are divisible by divisors from those which ai« not, and by
considering the iH^mainder that is left in tlie division of tlie
latter. For this purpose let us examine tlic divisors ;
2, 3, 4, 5, 6, r, 8, 9f 10, &c. ,
59. First, let the divisor be 2 ; the numbers divisible by it
aii3 £94,6, 8, 10, 12, 14, 16, 18^130| ficc. which, it appears
ebip.e. Of Bittij^ ttMiMHei. 17
Increase always hj tvo. Thaae nainberBy. as far as tfaejr can be
continaed, are called even namben. But there are other num-
bersy namely^
U 3, 5, r, 9, 11, IS, 15, 17, 19, &c.,
which are anifermly lessor greater than the former by unity,
and which cannot be divided by 2f without the rAnainder 1 ;
these are called odd numben.
The even numbers are all comprehended in the general expres-
sion ^a ; for they are all obtained by successively substituting
for a the integers 1, fi, 3, 4, 5, 6, 7, &c., and hence it follows that
the odd numbers are idl comprehended in the expression 2 a + If
because S a -f 1 is f^ater by unity than the even number 2 a.
60. In the second place, let the number 3 be the divisor ; the
numbers divisible by it are,
3, 6, 9, 12, 15, 18, 21, 24, 27, 30, and so on ; and these num*
bers may be represented by the expression Sa; for 3 a divided
by 3 gives the quotiient a without a remaindei^. All other num*
bers, which we would divide by 3, will give 1 or 2 for a remain-
der, and are consequently of two kinds. Those whichi after tho
division leave the remainder 1, are ;
1.4.7, 10,13, 16, 19, &c.,
»
and are contained in the expression 3 a -f 1 ; but the other kind,
where the numbers give the remainder 2» are ;
2.5.8, 11,14, 17,20,&a,
and they may be generally expressed by 3 a -f 2 : so that all
numbers may be expressed either by 3 a, or by Sa + 1, or by
3a + 2.
61. Let us now suppose that 4 is the divisor under considers*!
tion : the numbers which it divides are ;
4, 8, 12, 16, 20, 24, &c., ^
which increase uniformly by 4, and are comprehended in the
« expression 4 a. All other numbers, that is, those which are not
divisible by 4, may leave the remainder 1, or be greater than
the former by 1 : as
1, 5, 9, 13, 17, 21, 25, Ace,
and consequently may he comprehended in the expression
4 a + 1 : or they may give the remainder 2 ; as
2, e, 10, 14j 18^ 22, 26^ &c.,
EvL Mg. 3
18 MgOra. Sectl.
and be expressed by 4 a + 2 ; or^ lastly^ ibey may give tbe
remainder 3 ; as
3,7,11,15, 19, 23,27, &c.,
and may be represented by tbe expression 4 a -f 3.
All possible integral numbers are therefore contained in one or
other of thjdse four expressions ;
4a, 4a+l, 4a + 2, 4a + 3. r
62. It is nearly the same nvben the divisor is 5 ; for all nnm-
hers which can be, divided by it are comprehended in the
expl-ession 5 a, and tiu)se which cannot be divided by. 5, are
reducible to one of tbe loUowing expressions :
5a + l, 5a + %9 5a4-3, 5a-f4;
and we may go on in the same manner and consider the greatest
divisors.
63. It is proper to recollect here what has been already said
on the resolution of numbers into their simple factors i for every
number, among tbe factors of which is founds
2, or 3, or 4, or 5, or 7,
or any other nurobery will be divisible by those numbenf. For
example $ 60 being equal to2x2x3x5, itis evidant that 60
is divisible by 2, and by 3, and by 5.
64. Further, as the general e^Epression a 5 c d is not only divi^
sible by a, and fr, and c, and d, bat also by
aft, ac, aif ftc, ftd, cd, and by
aht, abd, acdf fted, and lastly by
abc d, that is tosay, its own value ;
it follows that 60, or 2 x 2 x 3 x 5, may be divided not oi4y by
these simple numbers, but also by those which are oonposed of
two of them ; that is to say, by 4, 6, 10, 15 : and also by those
which are composed of three of the simple factors, that is to say,
by 12, 20, 30, and lastly by 60 itself.
65. WheUf therefbre^ we have represented any number, assumed
at pleasure, by Us sxmjkfadtorsf it will be very easy to shew all
the numbers by which U is divisible. Far we have only, first, to
tahe the simple factors one by one, and tlien to multiply them togeth-
er two by two, three by three, four by four, ^c, till we arrive at
the number proposed.
06. It must here be particularly observed ; that every number
is divisible by 1 ; and also that every number is divisible by
Chap. r.
Of Simple ^lumtities.
19
itself; so (hat every number has at least two factors^ or divisors^
the number itself and unity ; but every number, which has no
other divisor than these two, belongs to the class of numbers,
which we have before called simple, or prime numbers.
All numbers, except these, have, beside unity and them-
selves, other divisors, as may be seen from the following table,
in which are placed under each number all its divisors.
TABLE.
1
ie
3
4
5
1
6
1
7
8
9
1
10
1
11
1
12
J
13
14
15
1
16
1
17
1
18
1
19
1
20
1
1
1
1
1
1
1
1
1
£
S
8
5
2
i
2
S
%
11
2
13
2
3
2
17
2
19
2
4
S
4
9
5
S
7
5
4
S
4
6
8
10
4
6
14
15
8
16
6
9
5
10
1
S
2
3
4
S
4
2
P.
12
6
«
4
5
18
2
20
6
S
4
2
2
4
2
6
p.
P.
P.
P.
P.
P.
P.
P.
»
67. Lastly, it ought to be obserfed that 0^ mr nothings may be
considered as a number which has the property of being divisi-
ble by all possible aumbers ; because by whatever number a
we divide 0, the quotient is always ; for it must be remarked
that the oniltiplication of any number by nothing produces noth-
ing, and-timrefore times a, or Oa^ is Ou
CHAPTER Vn.
OfFractUms in generals
68. Whsv a number, as 7 for instance, is said not to be
divisible by another number, Ibt us suppose by 3, this only
means, that the quotient cannot be expressed by an integral
number ; and it must not be thought by any means that it is
£0 JMg^a^ Sect 1.
impQBflible to form an idea of that quotieiit Onl j imagiito a
line of 7 feet in lengthy no one ican doubt the poseibili^ of
dividing this line into S equal partSf and of fomiog a notion of
the length of one of those parts*
69. Since therefore we maj form a precise idea of the quo-
tient obtained in similar caseSf though that quotient is not an
integral numbesTf this leads us to consider a particular species of
numbersy called fradumSf or ftrokcn mmben. The instance
adduced furnishes an illustration. If we have to divide 7 by S^
we easily conceive the quotient which should result, and express
i^ by 7 f piercing the divisor under the dividend, fuid separatin|;
the two numbers by a stroke, or line.
70. So, in genera{9 when the number ^ is to be diroided by (ft#
number b, we rq^resent the quatienit by -^ and eall. this firm (f
expression afraetim. We cannot therefore give a better idea of
a fraction -^, than by saying that we thus express the quotient
resulting from the division of the upper number by the lower.
We must remember alsoy that in all fractions the lower num-
ber is called the denominator, and that above the line the name"
rotor,
n. In the above fraction, }> which we read seven thirds f 7 is
the numerator, and 3 the denominator^ We must also read |» two
thirds ; |, three fourths ; |, three eighths ; -^^^^ twelve hun-
dredths J and |, one half.
72. In order to obtain a more perfect knowledge of the
nature of fractions, we shall begin by considering the case in
a
which the numerator is equal to the denooinator, as in—.
Now, since this expresses the quotient obtained by dividing a
by a, it is evident that this quotient is exactly unity, and that
consequently this fraction ^ is equal to 1, or one integer ; for
the same jreason, all the following fractions,
h h h h h h h *«.,
are equal to one another, each being equal to 1, or one integer.
73. We have seen that a fraction, whose numerator is equal to
the dcnominatort is equal to unity. All fractions therefore,
whose numerators are less than the denominators^ have a value
Cbap. 7. Of SmpU (ffumSiiM. %\
less tbau Qni<7'. For^ if I baTe a nmnber to be divided by ano«
ther wbich is greatery the result mast necessarily be less than 1 ;
if we cut a line, for example, two feet long, into three parts, one
of those parts will unquestionably be shorter than a foot : it is
evident then, that | is less than 1, for the same reason, that the
numerat(Mr 2 is less than the denominator 8.
74. tf the numerator, on the contrary, be greater than the
denominator, the Taloe of the fraction is greater than unity.
Thus I is greater than 1, for \ is equal to \ together with \.
Now \ is exactly t, consequently | is equal to 1 -f |, that is, to
an integer and a half. In the same manner ^ is equal to 14, f
to 14, and I to d|. And in general, it is sufficient in such cases
to divide the upper number by the lower, and to add to the
quotient a fraction having the remainder for the numerator, and
the divisor for the denominator. If the given fraction were, for
example, 44, we should have for the quotient 3, and 7 for
the remaindw; whence we conclude that 4| is the same as
75. Thus we see how fractions, whose numerators are greiiter
than the denominators, are resolTod into two parts ; one of
which is ah integer, and the other a fractional number, having-
the numerator less than the denominator. Such fractions as
contain one or more integers, are called improper JracHanSg to
dbtinguish them from fractions properly so called, which, hay-
ing the numerator less tlmn the denominator, are less than unity,
or than an integer.
76. The nature of fractions is frequently considered in an-
other way, wbidi may throw additional light on the subject.
If we consider, for example, the fraction .|, it is evident that it
is three times greater than |. Now this fraction ^ means, that
if we divide 1 into 4 equal parts, this will be the vdue of one of
those parts $ it is obvious then, that by taking 3 of those parts,
we shall have the Takie of the fraction |.
In the same manner we may consider every other fraction ;
for example, ^\ ; if we divide unity into 12 equal parts, 7 of
those parts will be equal to this fhiction.
77. From this manner of considering (IractionSf the expres-
sions numerator and iemminatar are derived. For. as in the
22 Algebra. Sect. 1.
preceding fraction /^i, tbe oumber under the line shews that 12
is the number of parts into which unity is to be divided ; and as
it may be said to denote, or name the parts^ it has not improper-
ly been balled tbe denominator.
Further, as the upper number, namely 7, shews that, in order
have the value of the fraction, we must take^ or collect 7 of
those parts, and therefore may be said to reckon, or number
them, it has been thought proper to call the number above the
line the wameraior.
78. As it is easy to understand what | is, when we know the
signification of ^, we may consider the fractions, whose nume-
rator is unity^ as the foundation of all others. Such are the
fractions^
1. 1. 1. 1 1 1 1 1. 1 X 1 Arn
T» T» T> T» T9 T> T> T» Tir» TT» Tl» ***'•»
Mid it is observable that these fractions go on continually dimin*
ishing ; for the more you divide an int^er, or the greater the
number of parts into which you distribute it, the less does each
of those parts become. Thus ^ ^^ is less than -^^ ; ^rVv ^ '^^^
than tJv ; and ^^\„ is less than ^^Vv
79. As we have seen, that the more we increase the denomi-
nator of such fractions^ tlie less their values become ; it may be
asked, whether it is not possible to make the denominator so
great, that the fraction shall be reduced to nothing ? I answer^
no ; for into whatever number of parts uni^ (the length of a
foot for instance) is divided ; let those parts be ever so small»
they win still preserve a certain magnitude, and therefore can
never be absolutely reduced to nothing.
80. It is true, if we divide the length of a foot into 1000 parts;
those parts will not easily fall under tbe cognizance of our
senses : but view them through a good microscq^e, and each of
them will appear large enough to be subdivided into 100 parts,
and more.
At present, however, we have nothing to do with what de-
pends on ourselves, or with what we are capable of performing,
and what our eyes can perceive ; the question is rather,' what is
possible in itself. And, in this sense of the word, it is certain,
that however great we suppose the denominator, the fraciien
win never entirely vanish, or become equal to 0.
Chap* 7. OfOmfU quanHUa. i»
81. We never therefore arrive completdj at nothing, how-
aver great the denominator may be ; and these fractions always
preserving a certain value, we may continue the series of
fractions in the 78th article without interruption. This circum*
stance has introduced the expression* that the denominator must
be infinite, or infinitely great, in order that the fraction may be
reduced to 0, or to nothing ; and the word infinite in reality
signifies here, that we should never arrive at the end of the
series of the above mentioned /rorfion^.
82. To express this idea, which is extremely well founded,
we. make use of the sign qd^ which consequently indicates a
number infinitely great; and we may therefore say ihatthis
fraction i is really nothing* for the very reason that a fraction
cannot be reduced to nothing, until the denominator has been
increased to infinity.
83. It is the more necessary to pay attention to this idea of
infinity* as it is derived from the first foundations of our know-
ledge* and as it will be of the greatest importance in the follow-
ing part of this treatise.
We may here deduce from it a few consequences, that are
extremely curious and worthy of attention. The fraction i
represents the quotient resulting from the division of the divi-
dend 1 by the divisor OD • Now we know that if we divide
the dividend 1 by the quotient i, which is equal to 0, we obtain
again the divisor qd : hence we acquire a new idea of infinity ;
we learn that it arises from the division of 1 by ; and we are
therefore entitled to say, that 1 divided by expresses a number
infinitely great, or QD •
84. It may be necessary also in this place to correct the
mistake of those who assert, that a number infinitely great is
not susceptible of increase. This opinion is inconsistent with
the just principles which we have lud down ; for ^ signifying a
number infinitely great, and ^ being in'contestably the double of
4f it is evident that a number^ though infinitely great, may still
become two or more times greater.
.;v
^ ' ' ' Sgdirai .• -Bwt t*
^
CHAPTER Vm.
C^ Hit fragcrtuti^
85. We have already aeeh, that each of the fractiottty
!• f i» T» ^ h {» *c-^
makes an integer, and that consequently they are all eqoal to
one another. The same equality exists in the following frac-
tionSf
f4 • fl 10* 1» %i.
» ¥> 7» Z» T » T » '**'•>
each of them making two inte^rs ; for the nnmerator of eacbf
divided by its denominatorf gives 2. So all die fractions
\> t» It V» V' V» &c.,
are equal to one another, since 3 is their common value.
86. We may likewise represent the value of any fraction, is
an infinite variety of ways. For if we mutttply Mh the nume^
rotor and the denominator rfa fraction by the same numbert which
may he aseumed at pleasure^ thts fracHon will etiU preserve tte
same value. For this reason all the fractions
h h h h re* A* A» tV» tV> ii* **^*»
are equal, the value of each being |. Also
h h h TJ* TT' Tt* TT» tV ¥T* ih *^»
are equal fractions, the value of each of which is ^» The frac-
tions.
^ J. tV ih a* Mf «• Ac.,
liave likewise aU the same value ; and lastly^ we may conclude
in general, that the fraction ^ may bo represented by the fol-
lowing expressions, each of which is equal to -ri namely^
a %a 3a 4a 5a 6a 7a .
7 W n* W 56' 6l' 7** *^*
87. To be convinced of this we have only to write for tlie
value of the fraction -7 a certain letter c, representing by this
letter c the quotient of the division of a by fr ,* and to recollect
that the multiplication of the quotient c by tiie divisor h must give
the dividend. For since c multiplied bj (gives a, it is evident that
c multiplied by 2 6 will give 2 a, that c multiplied by 3 5 will give
Chap. 6. Of Simple ^fuanliHes. SI5
S Of and that in general c multiplied by m b mmi give m a. Now
changing this into an example of division, and dividing the pro-
duct ma, by mb one of the factors, tlie quotient must be equal to
the other factor e; but m a divided by m 6 gives also the fraction
^f which is consequently equal to c ; and this is what was to
in b
he proved : for c having been assumed as the value of the frac-
tion 4fi^ is evident that this fraction is equal to the fraction
— rt whatever be the value of m.
mo
88. We have seen tliat every fraction may be represented in an
infinite fmmber qfformSf each of which contains the same value ;
and it is evident that of ail these forms, that, whicli shall be
composed of the least numbers, will be most easily understood.
For example^ we might substitute instead of | the following
fractions,
r9 T» T7» TT» Tf» "•*'•>
but of all these expressions | is that of which it ia easiest to
form an idea. Here therefore a problem arises, how a fraction,
such as -^^9 which is not expressed by the least possible numbers,
may be reduced to its simplest form, or to Us kast terms, that is
to say, in our present example, to 4*
89. It will be easy to resolve this problem, if we consider that
a fraction still preserves its value, when we multiply both its
terms, or its numerator and denominator, by the same number.
For from this it follows also, that if we divide the numerator and
denominator of a fraction by the same number, the fraction still
preserves tlie same value. This is made more evident by means
of the general expression — r ; for If we divide both the numc-
rator ma and the denominator m b by the number m, we obtain
the fraction — , which, as was before proved, is equal to — ^.
90. In order therefore to reduce a given fraction to its least
terms, it is required to find a number by wliich both the nunie«
rator and denominator may be divided. Such a number is
called a common divisor, and so long as we can find a common
divisor to tjie numerator and the denominator, it is certain that
the fraction may bo reduced to a lower form ; but, on the con-
S6 Algebra^ -Sect 1.
trary, when we see that except oniU no other common dinsor
can be foundy this shews that th^ffadion is already in the sim*
plest form that it admits of,
91. To make this more clear, let us consider tlie fractioii
Tiv ^® ^^ immediately that both the terms are divisible by
Sy and that there results the fraction |^. Then that it may
again be divided by 2, and reduced to 4^ ; and this also» having
2 for a common divisor, it is evident, may be reduced to -fj*
But now we easily perceive, that the numerator and denomina-
tor are still divisible by S ^ performing this division, therefore*
we obtain the fraction |, which is equal to the fraction proposed*
and gives the simplest expression to which it can be reduced }
for 2 and 5 have no common divisor but 1,' which cannot dimin«^
]sh these numbers any further*
92* This property of fractions preserving an invariable value*
whether we divide or multiply the numerator and denominator
by the same number, is of the greatest importance, and is the
principal foundation of the doctrine of fractions. For example*
we can scarcely add together two fractions, or subtract them
from each other, before we have, by means of this property,
reduced them to otiier forms, that is to say, to expressions whose
denominators are equal. -Of this we shall treat in the following
chapter.
95. We conclude the present by remarking, that all integers
mayalso be represented by fractions. For example, 6 is the
same as 4, because 6 divided by 1 makes 6 ; and we may, in the
same manner, express the number 6 by the fractions y , Y, '/,
V9 and an infinite number of others, which have the same value.
CHAPTER IX.
Of the Addition and Subtraction oj Fractions.
94. .When fractions have equal denominators, there is no
difficulty In adding and subtracting them ; for f -f- ^ is equal to
\f and ^ — I is equal to \. In this case, either for addition or
Chap. 9. Of SimpU (fuaniities. %7
anbtractkaif we alter only the numeratorsy and place the com-
mon denominator under the line ; thus,
TTrv + 'nnr"~TTrir'^Tinr-r TVT ^ equ^ xo -yi^^* tv^tt —
¥v + Ti is equal to 3 J, or II ; || — /^_|i +i| is equal to
||» or I ; also | +| is equal to |, or 1, that is to say, an inte-
ger ; and }-— I +7 is equal to I, that Is to say, nothing, or 0*
95* But wlien fractions have not equal denominators, we can
always change them into other frndiom that have the same denomi^
nator. For example, when it i^ proposed to add together the
fractions | and j, wq must consider that | is the same as |, and
that I is equivalent to } $ We have therefore, instead of the two
fractions proposed, these | + 1, the sum of which is {. If the
two fractions wei*e united by the sign minus, as | *^ I, we should
have I — f or |.
Another exaa^e : let the fractions' proposed be | + 4 ; since
4 is the same as |, this value may be substituted for it, and wo
may say | +f makes y , or 1 f, '
Suppose further, that the sum of | and | were required. I
•ay that it is ^\ ; for | makes ^\f and | makes ^.
96. fFe mofi have a greater number ^ fractimis to be reduced to a
eomnum denaminaior ; for example, |, f, |, |, | ; in this case
the whole depends on finding a number which may be divisible by
all the denommators of these fractions. In this instance 60 is the
number which has that property, and which consequently
becomes the common denominator. We shall therefore have
II instead of | ; || instead of | ; || instead of | ; || instead
of I ; and || instead of |. If now it be required to add together
all these fractions ||, ||« ||, ||, and ||, we have only to add
all the numerators, and under the sum place the common denomina"
tor 60 ; that is to say, we shall have \y , or three integers, and
ih ««• 3 ||.
9T» The whole of this operation consists, as we before stated^
in changing two fractions, whose denominators are unequal, into
two others, l^)fose denominators are equal* In order therefore
a c
to perform it generally, let -r and -^ be the fractions propos-
ed. First, multiply the two terms of th$ first fraction by d, we
•hall have the fraction r-^ equal to -r ; next multiply the two
S« Mgtbra. Sect 1.
tonus of the second fraction by 6, and we shall have an eqaiva-
bc
lent value of it expi-essed hy y-^ ; thus the two denoniinators
become equal. Now if the sum of the two proposed frac-
tions bo required, we may immediately answer that it is —y^ — ;
U (If
-J L
and if their diHcrcnce be asked, we say that it is — r-j — • If
bd
the fractions | and ^f for example, were proposed, we should
obtain in their stead ^| and |f ^ of which the sum is \^^j and
the difference 44.
98. To this part of the subject belongs also the question, of
two proposed fractions, which is the greater or the less ; for, to
resolve this, we have only to reduce the two fractions to the
same denominator. Let us take, for example, the two fractions
I and 4 : when reduced to the same denominator, the first be-
comes 1^, and the second ^4' ^'^^ ^^ i^ evident that the second,
or {-. Ls the greater, and exceeds^ the former by 7^^^*'
Again, let the two fraction ^ and 4 be proposed. We shall
liave to substitute for them, \^ and |4> ; whence we may conclude
that 4 exceeds |, but only by ^V*
99. When it is required to subtract afradion from an integerf
it is sufficient to change one of the units of that integer into afraC'
Hon having the same denominator as the fraction to be subtract^
ed ; in the rest of the o|)eration there is no diflSculty. If it
be required, for example, to subtract | from 1, we write | in-
stead of 1, and say that | taken from | leaves the remainder 4.
So ^'7, subtracted from 1, leaves •/,.
If it wei*e required to subtract ^ from 2, we should write 1
and ^ instead of 2, and we should immediately see that after the
subtraction tliere must remain 1 ^.
100. It happens also sometimes, that having added two or
more fractions together, we obtain more than an integer ; that
is to say, a numerator gi*eater than the denominator : this is a
case whicli has already occurred, and deserves attention.
We found, for example, article 96, that the sum of the five
fractions |, f, |« 4, and 4, was Vt , and we remarked that the
value of this sum was^S integers and |J, or |^. likewise | -f
h 0** Tf + t\> niakes 45, or l-/^. We Jiave only to perform the
Chap. 10. Of ample quantUies^ 29
actaal division of the nnmerator by the denominatorf to see how
many integers there are for the quotient, and to set down the
remainder. Nearly the same must be done to add together
numbers compounded of integers and fractions ; we first add
the fractions, and if their sum produces one or more integers^
these are added to the other integers. Let it be proposed, fur ex-
ample, to add S^ and ^ ; we first take the sum of ^ and |, or of
^ and |. It is j. or 1| ; then the sum total is 6|.
CHAPTER X.
Of the JUuUipliealiim and Bividon of Fractions*
lot. The rule for the mnUipUcatum of a fraction by an integer,
or whok numberf is to muUiply the numerator only by the given
number, and not to change the denominator : thus, ^^
S times, or twice } makes |, or 1 integer $
£ times, or twice 4 makes | ;
S times, or thrice j- makes |>« or| ; and
4 times -/^ makes 4^ ^^ Wi* ^^ ^*
Bui, instead of this rtde, we may use that of dividing the denom*
inator by (he given integer ; and this is preferable, when it can be
used, because it shortens the operation. Let it be required, for
example, to multiply f by 3 ; if we multiply the numerator by
the given integer we obtain y , which product we roust reduce
to |. But if we do not change the numerator, and divide the
denominator by the integer, we find immediately |« or S | for
tlie given product Likewise || multiplied by 6 gives y , or 3^.
102. In general, therefore, the product of the multiplication
of a fraction -7 by c is -^ ; and it may be remarked, when the
integer is exactly equal to Hie denominator, that the product must
he equal to the numeratorm
C^ taken twice gives 1 ;
So that < I taken thrice gives 2 ;
L| taken 4 times gives 3.
And in general, if we multiply the fraction -r by the number
b, the product must be o^ as we have already shewn ; for since
so 4i!f06f«» BeKLU
4 expresses the quotient resulting from the division of the divi-
o
dend a by the divisor b, and since It has been demonstrated that
the quotient multiplied by the divisor will give the dividend^ it
18 evident that -r multiplied by b must produce a.
103. We have shewn how a fraction is to be muItipKed by an
integer; let us now consider also how a fraction is to be droided
ty an integer $ this inquiry is necessary bofore we proceed to the
multiplication of fractions by fractions. It is evident^ if I have
to divide the fraction | by 2, t^at the result must be | ; and
that the quotient of | divided by 3 ia 4. The rule therefore is,
to divide the numerator by the integer Toithout changing the de»
nominator. Thus,
II divided by d gives /^ ;
\\ divided by 3 gives ,\ ; and
II divided by 4 gives ^\ ; &c.
104. This rule may be easily practised, provided the nume-
rator be divisible by the number proposed ; but very often it is
not : it must therefore be observed that a fraction may be trans-
formed into an infinite number of other expressions, and in that
number there must be some by which the numerator might be
divided by the given integer. If it were required, for example,
to divide | by 2, we should change the fraction into !« and then
dividing the numerator by S, we should immediately have | for
the quotient sought.
In general, if it be proposed to divide the fraction -^ by c, we
change It Into ^, and then dividing the numerator a c by c,
write -r- for the quotient sought.
c
105. When thertfore a fraction ^istobe dvrided by an integer
c, we have only to mvUiply the denominator by that number^ and
leme the numerator as it is. Thus | divided by 3 gives /^, and
I divided by 5 gives ^%.
This operation becomes easier when the numerator itself is
divisible by the integer, as we have supposed in article 103.
• f
«■>
Chap. 10. ^ Of Bimph ifuantUies^ SI
For example, yV divided by 3 would give^ according to our last
rule, 7T ' ^"^ ^y ^^^ ^"^ ^^^^* which is applicable here, we
obtain ^V^ an expression equivalent to ^'^ , but mora simple.
106. We shall now be able to understand how one fraction --r
may be multiplied by another fraction -^^ We have only to
c
consider that -^ maans that c is divided by d; and on this prin*
ciple, we shall first multiply the fraction -? by c, which pro-
c
duces the result -r- ; after which we shall divide by d, which
• 6 -mU*
J9inice ihe foOowing rvU for multiplying fractions ; mvlHplif
separatdy the fiumerators and the denominators. ^
Thus i by I gives the product f , or ^ ;
• by I makes ^V y
I by ^ produces ^9 or j'j ; &c.
lOr* It remains to shew how one fraction may he divided by
another, ^e remark first, that if the two fractions have the same
number for a denominator, the division takes place only with
respect to the numerators ; for it is evident, tliat -^^ is contain*
ed as many times in y\ as S in 9, that is to say, thrice ^ and in
the same manner, in order to divide ^^ by ^^^ we have only to
divide 8 by 9, which gives |. We shall also have ^^ in 4^, 3
times : , J^ in ^^^, 7 tiroes ; ^^ in /y, 4. ; &c.
108. But when the fractions have not equal denominatorSf we
must have recourse to the method already mentioned for redac-
ing them to a common denominator. Let there be, for exam-
a c
pie, the fraction — to be divided by the fraction ^; we first re-
n. d
duce them to the same denominator ; we have then 7-- to be
d
b c
divided hy r^ ; it is now evident, that tlie quotient must be
represented simply by the divisiqn of a d by ( c ; which gives t-»
SS • Mi^iu, Sect f
Hence the following role : JUidtiply the numerator of the dvou
iend by the AenamiiuUor of the divisorf and the denominator of the
dividend by the numerator of the divisor ; the firet yroduet imll be
the numerator oj Uie fuatientf and the second will be its cfenomt*
natar»
109. Applying this rule to the division of | by !» we shall
have the quotient ^f ; the division of | by | will give } or | or
I and I ; and || by | will give ^ J« , or |.
110. This rule for division is often represented in a manner
more easily remeoiberedy as follows : Invert the fraction which
is the divisor^ so that the denominutar may be in the place of the
numerator, and the latter be written under the line ; then mulHply
the fractionf which is the dividend by this inverted fraction, and
the product wiU be Vie quotient sought. Thus l divided by | is
the same as | multiplied by ^9 which makes ^9 or 1 1. Also 4
divided by | is the same as | multiplied by f, which is \^ ; or
II divided by | gives tiie same || multiplied by |9 the product
of which is |f^^ or|.
We see then, in general, that to divide by the fraction !» is the
same as to multiply by 4» or 2 ; that division by | amounts to mul-
HpUcation by ^, or by S, j'C.
111. The number 100 divided by ^ will give 200 ; and lOOa
divided | will give SOOO. Further, if it were required to divide
1 by f ^Vtt^ ^'^^ quotient would be 1000 ; and dividing 1 by
Tvinrw ^^^ quotient is 100000. Tliis enables us to conceive
that, when any number is divided by 0, the result must be a
number infinitely great ; for even the division of 1 by the small
fraction ^^^^^^^j^^j^ gives for the quotient the very great num-
ber 1000000000.
112. Every number when divided by itself producing unit j^,
it is evident that a fraction divided by itself roust also give 1 for
the quo^tient. The same follows from our rule : for, in order to
divide * by |, we must n)ultiply | by ^, and we obtain ^ff or 1 ;
and if it be required to divide -y by -?• we multiply — by — ;
now the product — is equal to 1.
11.
<y SEoipk ^ttonftfies.
dS
115, We have still tomdain au expresBion whick is^ fre-
qoeiitiy used. It OMiy i)e askedj foi* exan^^ wbajt is. the half
of. 1 1 this means thai w^ must nmltiply i by ^« So likewise^ if
the value of | of 4 wsra r^W'^^^ ^P should multi^y | by ^y
which produces ^^ ; and | of -^^ is the same as -^V mtdtipiied by
|# which prodiices f ^
1 14* Lastly, ws mmst here observe the same rules with respect
to the signii + an4»-r> thj4 wo b^fiorelaid down for integers^
Thus + 1 multiplied by —^^ makes — }} and ^ | multipled by
-^ 4 gives + ^. FartfaelTy -^ f divided by + 1 makes ~ || ;
and <— j divided by— i n^akes -f 41 ^r + !•
CHAPTER XQ
. Of Square ^mbers.
/ 115. The product tf a number p when multiplied by itsdf, U
called a equare; and for this reason, the number, considered in
relaHon to such a product, is caUed a square root.
For example, when we multiply 18 by 12, the product 144 is
a square, of which the root is 13,
This term is derived from geometry, which teaches us that
the contents of a square are found by multiplying its side by
itself.
116* Square numbers are found therefore by multiplication ^
that is to say« by multiplying the root by itself. Thus 1 is the
square of 1, since 1 multiplied by 1 makes 1 ; likewise, 4 is the
square of 2 ; and 9 tlie square of 3 ; 2 also is the root of 4, and
3 is the root of 9.
We siiall begin by considering the squares of natural numbers,
and shall fii*st give the following small table* on the first line of
which several numbers, or roots, are placed, and on the second
their squares.
Numbers.
Squares.
I
1
2
4
3
9
4
5
^5
6
36
7
4<>
8
64
9
81
10
100
11
121
12
144
13
169
IM.Mg.
34
Mgebra.
SectU^
117. It will be reodilj perceived, tHat the seHes of aqtiar^
irambers tliua arranged has a Bingular property ; namelyt that
if each of them be subtracted from that which immediately
fallows, the remainders always increase by £« and form thia
aeries;
3, 5, 7f 9f Uf 13, 15, 17, 19, !il» &c.
1 1 80 The squares of fracHoHs are found in the same fRmuter, by
mvUiplying any gvvenfrQcHon hy tts^. For exam^e^ the square
The square ot^ ^ J>is
of^
We have only therefore to divide the square of the numerator
by the square of the denominator, and the fraction, which ex-
presses that division, most be the square of the given fraction.'
Thus, If is the s^are of f ; and reciprocally, f is the root
01 Tx*
119. When the square of a mixed number, or a immbert com«
posed of an integer and a fraction, is required, we have only to
reduce it to a single fraction, and tlien to take the square of thi^t
fraction. Let it be required, for example, to find the square 6f
£| ; we first express this number by |, and taking the square
of that fraction, we have y , or 6^, for the value pf the square
of 2|. So to obtain the square of 3^, we say 3^ is equal to y ;
therefore its square is equal to y/, or to 10 and ^Y* The
squares of the numbers between 3 and 4, supposing them to
increase by one fourth, are as follows :
!N umbers.
SqiMns.
3
9
Si
IOtV
4
16
From this small table we may infer, that if a root contain a
fraction, its square also contains one. Let the root, for example,
be 1/; ; its square is 44I, or S^f^ ; that is to say, a little great-
er tiian the integer 2.
120. Let us proceed to general expressions. When the root
Is a, the square must be a a; if the root be 2 a, the square is 4 a a;
;
\
Chap* IS. ^ Btn^ie ^mntUies. d^
^bkh shews that hy doiibling the root^ the square becomes 4
limes greater. So if the root be 3 a» the square is 9 a a; and if
•the root be 4 a, the square is 16aa. But if the root be fi6, the
square isaabb; andif the root beat Cf the square haabbce*
121. Thus ToAen the root is composed ^ two, or more JdetarSf
we muUiphf their squares iogdher; and reaprooaUg, if a square
be composed of two or more faeterSf ef which eaA is a square, we
tove only to mmUipiytogeth^ tiie roots of those squares, to obtain
the eofm^^Me root t^ihe square proposed. Thus» as £S04 is equal
to4xl6xS69 tlie square root of it is 2x4x6>or48; and
48 is found to be the true square root of 2304, because 48 x 48
gives 2304.
122* Let us now consider what rule is to be observed
with regard to the signs + and ^-v First, it is evident that
if. the root'has^ tlie sign -ff that is to say, is a positive num-
ber, its sqimre must necessarily be a positive number also»
because + by + makes -f^ the square of +a will be+aa«
But if the root be a negative number, as — a, the squaro is still
positive^ for it is 'i-aa; we may therefore conclude, that -fa a
is the square both ^ +a aad ^— a^ and that consequently every
square has two roots^ one positive and the other negative. The
square root of 25, for example, is both •+> 5 and — 5, because
— 5 multiplied by r-«- 5 gives 25^ as well as + 5 by -f 5.
X-
CHAPTER Xm
Of Square Boots, and of TrraHotud J^Tunbers resulting from them^
123. What we have said in the preceding chapter is chiefly
this : that the square root of a given number is m^tliing but a
number whose square is equal to the given number; and that
we may put before these roots either the positive or the negative
sign.
124. So thai when a square number is given, provided we
retain in our memory a suflBlcient number of square numbers, it
is easy to find its root. If 196, for example^ be the given num*
her, we know that its square root Is 14.
SB Jlgdmk/ Stet 1.
Fractions Hkewfi^e are ed^ily iiinaged : it 19 eviieat^ far
example, that 4 fs the square raot of ff • To be. OtavwceA of
this, we have only to take the square root of the MHnetfator^ m4
that of the denominator.
If the number proposed be a mixed number, as lft|» We redoce
jt to a single fraction, which here is Y, and we iiiunediateiy
perceive that I, or S |, must be the square root of 152^*
125. But when the given number is liot a square, at iS, lir
example, it is not iJbssible to extract i^ square root t ortolM
a number, wbicii, jnidtiplied by itself, will give the product ISU •
We know, however, that the square root of IS must be greater
than 3, because 3x3 produces only 9 : and less than 4, beeanse
4x4 produces 16, which id mOre than 12. We Icndw also, that
this root is less than 8 ^ ; for we have seen that tlie aqnare of
3 1, or I is 12 1. Lastly, we may approach litttl neioier to this
root, by comparing it with S^; fot the square of S ^, or of ^
^^ ViV' ^^ ^^7TT> ^^ that this fraction is stOl j^reateT tha»lho
root required ; but very little greater, as the diflbrence of tiio
two sqares is only ^|y.
126. We may suppose that'as^S | and 3 -{^ are numbers gri^at^
than the root of 12, it might be possible to add to -8 a fraction
a little less than ^^y, and precisely such that the square of the
Bum would be equal to 12.
Let us therefore try with 3^, since ^ is a little less than -j^y.
Now 34 is equal to y, the square of which is y/, and conse«
quentiy less by ^ than 12, which may be expressed by Y/ .
It is therefore proved that 3^ is le^, and that 3^\ is greater
than the root required. Let us then try a number a little gi*eater
than 34, but yet less than 3^^^, for example, 3^'^. This number,
which is equal to 44» ^^^ for its sqt^are VrV* Now, by reduc-
ing 12 to this denominator, we obtain \Y/ > which shows that
S^\ is still less than the root of 1 2, viz. by ^1^. Let us there-
fore substitute for -/^ the fraction -^^9 which is a little greater,
and see what will be the result of the comparison of the square of
S-^-y with the proposed number 12« The square of S^-^ is ^/^ ;
now 12 reduced to the same denominator is YtV 9 ^^ ^^ Vt ^
still too small^ though only by yf^, whilst 3 ^\ has been found
too great.
/
19,7 • It is evident tteFefore^^al whatever friiction be jeined
te S» tlie square of that anm must always centain a fraction, an4
can never be exactly equal to the integer 12». Thu8» although
we know that the square root of 12 is greater than 3 ^ and lesf
ttmn^ ^9 yet we areunaUeto assiga^n intermediate fractioii
between these twoj which^ at the same time, if added to s^ would
express exactly the square root of 1£« Notwithstanding this,
we,are not to assevt (bi^t the square root of 1£ is absolutely and
in iiaelf indeterminate ; it only follows from what has been said,
that this root, though it necessarily has a determinate 0ii^ni«
tHrie, cannot be expressed by fractions*
l&S. Th&re is therefore a sort of ^n^ers tvhkh cannot he
mas^gned kif JractionSf and ivhkh are nevertheless determinate
quanHties^ the square root of 12 furnishes an example. We
tall this new species of number^, irrational nundfers ; they occur
whenever we endeavour to find the squai^ root of a number
which is not a square. Thus, % not.heing a perfect square, the
equare root of 2,,or the number which, multiplied by itself,
would produce 2, is an irrational quantity. These numbers are
dso called snrd qikointUieSf or inannin^isuf oi/e^.
Ifi9. These irrational quantities, though they cannot be ex-
pressed by fractions, are nevertheless magnitudes, of which we
may form an accurate idea. For however concealed the square
root of 12, for example, may appear, we are not ignorant, tliat it
must be a number which, when multiplied by itself, would
exactly produce 1£ ; and this property is sufficient to give us an
idea of the number^ since it is in our power to approximate its
value continually. <
ISO. As we are therefore sufficiently acquainted with the nature
of the irrational numbers, under our present consideration, a par-
ticular sign has been agreed on, to express tlie square roots of all
numbers ithat are not perfect squares. This sign is written
thus v ^^ is ^^^d square root. Thus, vis represents the
square root of 12, or the number which, multiplied by itself,
produces 12. So, \/2" represents the square root of 2 ; V3"that
Of 3 ; vl that of 4 and, in general, \/T represents the square
root oj the number a. Whenever therefore we Would express the
C3
58 JKgOra* Sect U
flqnare root of a mmtber whicb is not ft flquMrty we need onlf
make use of the mark V ^J placing it before tke namber.
131. The eiplanaliony which we bave^ven of irrational num*
berst will readily enable us to apply to them the known meflKidiv
of calculation. For knowing that the aqoare root of d» multi-
plied by itself, must produce 12 ; we know also, that the multipli-
cation v^ by \/T most necessarily produced ; that, in tfae^same
manner, the multiplication of vf by s/T must give 3 : that x/T
by \/T makes 5 ; that ^/| by Vt makes | y and, in general^ that
V^rfmrfttpbed fty y r produces a.
13S. But when it is required to mtiU^y ViT iy x^'h' the produd
will he found to he Vab , becalise we have shewn before, that if a
square has two or more factors, its root must be composed of
the roots of those factors* Wherefore we find the square root
of the product a &, which is \/7S, by multiplying the square root
of a or \/7, by the square root of & or v^ It is ertdent from
this, that if b were equal to a, we should have x/Ta for the pro-
duct of Vo" by v^ Now ^/ilTa is evidently a, suice an is the
square of a.
133« In division^ if it were required to divide \/^ fbr exam-
ple, by ^b^ we obtain \y ; and in this instance the irration-
ality may vanish in the quotient. Thus, having to divide v^ii
by \/sl the quotient is vV*, ^bich is reduced to \/t» <uid conse-
quently to |, because ^ is the square of |«
134. When the number, before which we have placed the
radical sign v, is itself a square, its root is expi'essod in the usual
way* Thus v^ is the same as 3 ; ^9" the same as 3 ; y^ss the
same as 6 ; and \/ii^ the same as l^ or 3|. In these instances
the irrationality is only apparent, and vanishes of course.
135* It is easy also to multiply irrational numbera by ordi-
nary numbers* For example, 2 mnltiplied by ^T nu^es 2 vJT
and 3 times x/T make 3 y^ In the second example, however,
as 3 is equal to viT ^^ ^^7 ^i^e express 3 times vi" by \/9^
times v% ^^ by vl8* So 2 v^ is the same as V^a, and 3 x/tT
tlie same as \/9a* And, in general, b x/a' has the same viUue
as the square root nf b b a, or v abb $ w hence we infer recipro-
eallyi that when the number which is preceded by the radical 'v'
<*^
Chap* tt«
Of Simple fuanKUes.
S9
•ign contain* a scpiavey w^ may take tlie root dF fhat square and
put it before the sign, as we ghouhi do in writing b %/^ instead
of \f7Tb* After thia^ the fbUowing reductiims lirill be easily
uoderBtood : ^
>is equal tof<
VbT or V2'4
VTsT or v'm"
Vs^T or v5T
V^iT or v'frB
vfST or VS^
and 80 on«
136. Division is founded on the same principles*
^ V^ gi^^ ^9 or \^ In the same mannert
'2 V3 5
S V^J
4 VFi
.5 VS"J
V8
Via
7;;|= >is equal to<!
j-f orv4> oraj
Further
vr
2 ^
V2"
\ 2
f or ^9, or 3 ;
-= ^is equal to<
12
V6"
dirideif
, or V4, or 2,
~z9 orv4»^**V2 ;
^f or V7» or vsT;
^^, orv^JSoryS*
Lv6
or Vfr^t or lastly 2 ve"*
137. There is nothing in particular to be observed with
respect to the addition and subtraction of such quatitities, be-
cause Me only connect them by the signs + and — . For
^xample^ vF odded to x/T is written ^ F + v3" } and \^T sub-
acted from v^r M written v5"— V^T
-<^
n
40 JBgAra. Boot 1«
138. We BMqr obMrve laetly, titat in order to dietmgaiab irra-
tional iMiinbera» «e call all otbernuBibera^betiiintogral and frao*
tiomlf raiiamai mmberi. .
So that» wbeqiBver we speak of rational numbers^ we under-
stand integers or fractions*
CIUPTEKXXIIL
Of Impossible or Imaginary QuanHties, which arise from the
same source.
139. Wk have already seen that the squares of nnnbers*
negative as well as positive, are always positive^ or affected
with the sign -I-; having shewn that— -a midtiplied by— «a
gives -I- aa, the same as the prodaot of +a hy -fa. WherefiM%
in the preceding chapter* we supposed that all the nambersf of
which it was required to w^tract the square roots, were positive^
140. When it is required therefore ta extract the root of a
negative numberf a vecy great diiiculty arises $ since there is
no assignable number, the square of which would be a negative
quantity* Suppose, for example that we wished to extract the
root of — 4 ; we require such a number, as when multiplied by
itself, would produce — 4 ; now this number is neither + £ nor
— 2, because the square, both of + 2 and of — 2^ is +4, and not
— 4.
141. We must therefore conclude, that the square root of a
negative number cannot be dtiier a posi^ve numberf or a negative
mmberf since the squares of negative numbers also take the sign
plus. Consequently the root in question must belong to an en-
tirely distinct species of numbers ; since it cannot be ranked
either among positive, or among negative numbers.
142. Now, we before remarked, that positive numbers are all
greater than nothing, or 0, and that negative numbers are all less
than nothing, or ; so that whatever exceeds 0, is expressed by
positive numbers, and whatever is less than 0, is expressed by
negative numbers. The square roots of negative numbers,
therefore, are neither greater nor less than nothing. We can*
Chip. 18. €f amfk^ft^mHtU$. 41
ii<it say hiiwever, Aat they are' ; for moHipIied by pro-
diices 0, and consequently does not give a negative mimber.
143. Now, since aU numbeps^ whith it is possible to conceive^
fltre either grater or less than 0, or are itself, it is evident
that* we cannot rank the square ^ root of a negative number
amongst possiible numbers, and we must therefore say that it is
an impossible quantity. In this manner we are led to the idea
of numbers which from their nature are impossible. These nutii-
hers are umtdhf called ma^nary quaiMUeSf because they exist
Biierely in the .imagination.
144. All such expressions, as v^l) V^2, V^3, V-^. ^•f
ave consequently impossible, or imaginary numbers, since they
represent mij^ ts of negative qaantities : and of such numbers we
may truly assert, that they are^neither nothing, nor greater than
nothing* nor less than nothing $ which necessarily constitutes
HMD imaginflry, or impossible.
145. Btt« aetwilbstaiidingi all this, these numbers present
themselve8> to the mind 9 they exist in oor4magination» and wo
still have a •eafflcient idea of then ; since we know that byv'Z^
is meant' a mimber whieh, mult^ied by itself, produces — 4.
For this nsafeon also, notUng prevents us from making use of
these imaginary naiabersr and emplo^yiog them in calculation.
146. The first idea that occurs on the present subject is, that
tlie aqaave of v^> fo>^ example, or the product of x/Zls by
%/i^3^ must be^— 3 ; that the product of v^l by v^i is — 1 ;
and, in general,- that by multi^ying \/^ by V— a* or by taking
the square of \Ala* ^^ obtain — a.
• 147«Now, as — a is equal to + a mnlti^ied by — 1, and as
the square root of a product is found by multiplyii^ together
flie roots of its fiw;tors» it follows that the root of a multi-
plied by — V or %/^ is equal to \/r multiplied by \/Z.u
Now vT is a possible or real number, consequently the whole
impoeeiUtihf rf an imaginarff qtumiUy may be always reduced to
%^IZi, For this roason, vZ4 is equal to V4 multiplied by
V^l, and equal to 2 \A^1> on account of x/i" being equal to 2.
For the same reason, v-^9 is reduced to s/T X \/^l, or 3 \/^ij
ud v^^^ is ^ual to 4 V— 1.
EuU Mg. 6
fn
42 JUgOra. StaLU
148. Moreover, as \/T multiplied by \/T makes voSy ve
shall have x/F for value of ^I^ muUiplied by ^/^ ; and v^ or
2, for the value of the product of v^Hi by vHi. We see, tbM«»
fore, that two imaginary numbers^ muUiplitd together, produce a
realf or possUfle one.
But, on the contrary, a possible number, multiplied by an tut-
possible number f gives ahoays an imaginary product : thus, yds
hy v'+5 gives %/^^^
149. It is the same with regard to division ; for \/a divided
by s/V making 1^, it is evident that \/^ divided by v^i ^«ffl
Sb ,
make v-H* or 2 ; that v+a divided by ^z:3 will give
and that 1 divided hy v^l gives flLLf or V— 1 9 t^ecause 1 is
equal to v/4-u
150. We have before observed, tliat the square root of any
number has always two Values, one positive and the other
negative ; that V'C for example, is bdth -|- S and — 3, and thi^
in general, we must take — x/a as well as +\/a' for the square
root of a. This remark applies also to imaginary numbers ;
the square root of *^Kis both + yCIa and — x/^ ; but we must
not confound the signs + and — , wkiA are befbre the tadiealiB^
\/t wHh the sign which comes after it.
^ 151. It remains for us to remove any doubt wMck misyht
.-entertained concerning the utility of the numbers of which we
have been speaking ; for those numbers being impossible, it
would not be surprising if any one should think them entirely
useless, and the subject only of idle speculation. This however
is not the case. The calculation of imaginary quantities is of the
greatest importance: questions fluently arise, of which we
cannot immediately say, whether they include any thing real
Slid possible, or not. Now, when the solution of such a ques-
tion leads to imaginary numbers, we are certain that what is
required is impossible.*
* This is followed in the original by an example intended to illustrate what
is here said. It h omitted by the Editor, as it implies a degree of acquaint,
ance with the subject, which this learner cannot be supposed to possess at this
stage of his progress. \ r
I X
*
Cta^ 14*
Of BifttfU ^MnhHfV
43
CHAPTER XIV.
<2f CuHc Mmbers. ""
152. When a number has been multiplied iroice fty itself ^ or,
tvhich is (he same thingf when the square of a number has been
multiplied once more by that numbeff we obftdn a produd which is
called a cuhCf or a cubic number. Thus, the cube of a is a a (I9 since
it is the product obtained by multiplying a by itself, or by a, and
that square aa t^in by a.
The cubes of the natural * numbers therefore succeed each
other in the foUowin)^ order.
Numbers.
Cubes.
1
1
3
27
4
64
5
125
6
216
7
343
B
512
9
729
10
1000
1S3. U we consider the differences of these cubes, as we
did those of the squares, by subtracting each cube from that
vhich comes after it, we shall obtain the folio wing series of num-
bers ;
7, 19, Sr, 61, 91, 127, 169, 217, 271.
At first we do not observe any regularity in them ; but if we
take the respective differencea of these numbers, we find the
following s^ies :
12, 18, 24, 30, 36, 42, 48, 54, 60;
fai idiich the terms, it is evident, increase always by 6.
154* After the definition we have given of a cube, it will not
be diflBicult to find the cube of fractional numbers ; 4 is the cube
of ^ ; YT 1^ ^^ ^^^ ^^ ¥ f ^^^ XT ^ ^^ ^"^® ^^1* I*^ ^^
same manner, we have only to take the cube of the numerator
and that of the denominator separately, and we shall ha^e as
the cube of |, for Instance, \\m
155. If it be required iojini the cube ijf a mixed number^ we musl
first reduce it to a single fraction^ and then proceed in the manner
that has been described. To find, for example, the cube of 1^,
we must take that of |, which is '/, or 3 and |. So the cube
of .1^. or of the single fraction |, is yi/y or l|i ; and the cubd
of S J, or of V >8 *H% or 34|i.
44 Slt*^. tSptt^
156. Shiceaaaii tfaecttbeoftf^ tlMit«rol wiilbeaaaftdft/
whence we 8ee# that t^ a member im$ tmo or man factors, we
mayjind Us aihe by mMijifins together iheeiAes ef thoee fadmrSm
For example, as 12 is equal to S X 4, we multiply the cube of S,
which 18 £7, by the cube ^F 4» wkkh is 64^ and we obtain 175L^$
for the cube of IS. Further, the cube of 2 a is 8 a a Of and cona^
quently 8 « times greater than the cube, of a •- and like wise^ the
cube of da is £r a an, that is to say^sr times greater than the
cube of a.
157. Let us attend here also io the rigns + a«f-^. It is
evident that the cube of a positive 'nuinW + a moat atoo be
posit i ve, that is -f a a a. But if it be recjuired to cobe » negative
number —a, it is found by 4lrsl taking tbe -square, wMch 48
+ aaf and tlien multiplying, according to the rule, this square
by — a, which gives for the ciAe required — daiu lift this
respect f therefore, it is nx^ the same ntfith cubic numbers as wUh
squareSf since the latter ate ahoays pasiHve : ivhereao the coke
of _ 1 is —1, thai of'^lt is^6y that of ^ Sis ^^9 and
so on*
CHAPTEjaXV.
Of Cube BootSf and of irrational wimbers restdHngfrom them.
158. As we can^ in the nvanner already explained,- find tl^e
cube of a given number, so, when a number is proposed, we may
also reciprocally find a number, which, multiplied twice by itself,
will produce th^ number. Tbe number here sought is c^lsi|
with relation to the other, the ciAe rocL So thai thtffl(//li0H^
a gi^en number is tfte number whose cube is efoal to that gpocn
numoer.
15d. It is easy therefore to determine the cube root, when the
number proposed is a real cube^ sacb as the examples in tbe last
chapter* For we easily perceive that the cube root of 1 is 1 ;
that of 8 is 2 ; that of 27 is S ; ttiat of 64 is 4, and so on. And
in the same manner, the cube root of — *2r is-^d; and that of
•—125 is — 5. yj^J
Cktf.tS. Of OmflBl^iunMies. 46
Fitvib«r>tf ^ite |»opMe4 umber lie iiDnetkiiiy aa /f« the cube
root of it ooHst be | ; and that of i,^ is 4. LaaUyt the cube
soot af a fliicad aiHaber a^^ laost be 4^'or ij^ : beoaitae ^^ is
eqoai to f^.
160. But if the proposed number be not a cube, its cube root
cannot be eiq^reflsed dMiei* in iati^rs» ar ia fractional num-
bers. For exampley 45 is not a cubic number ; I aagf there-
fore that it is impossible to assign any number^ either integer
or fractional^ whose cube shall be exactly 45. We may haw-
ever aflkm^ that the cube root of that number is greater than
at since the ^ube of d is iHily 97 ; Md lass .than 4^ bei^use
the cube of 4 is 64. We know therefore, that the cube root
required is necessarily contained between the nambers d
and 4.
16U Since the cube root of 43 is gi«ater than S, if we add a
fraction to 3, it is certain that we may approximate still nearer
and nearer to the true value of this root : but we can never
aasign Ihe number which eiqprasees that value exactly ; because
Hbfd cube of a mixed nnmber can nev^ be pecfeclly equal to an
jutoger, such as 43. If we weise; to aappos^ for example, 3|, or
^ to be the cube root requiFe49 the error would be | ; for the
cube of I is only ^|', or 42|.
162. This therefore shews, that the cak rust of 43 carm0t Jk
expressed in enjf 'wmi/t miher btf integers in* by fraettans. How-
aver we ha^e a distinct idea of the magnitude of this root $
9
which induces us to use, in order to represent it, the sign x/r
which we place before the proposed number, and which is read
euheroat^ to distinguish Ufrom (he square reoi, which is often called
simply the rccL Thus V43 means the cube root of 43, that is to
say, the numbor whose cube is 43^ or which, multiplied twice
by itself, produces 43.
163. It is evident also, that such expressions cannot belong,
to rational quaatities, and that they rather form a particular
species of irrational quantities. They have nothing in common
with square roots, and it is not possible to express such a cube
root by a square root ; as^ for e:wDple, by x/'Ji ; for the square
46 Jtgikm^ Seeb f «
^f \/T2 being tS» its calie will be Id vi^f cims^jqaently still irra-
tional, and such cannot bo equal to 4d. »
164. If tbe projiosed tiuuber be a real cabe, our expressions
become rational ; x/i is equal to 1 ; V8 is equal to S ; ^27 is
s
equQl to S ; and, gpimvVtff Vm a« U eqmi ioM.
a
165. If it were proposed to nivUiply one cube root, v/a» ly atiatherf
y^, the product tnusf be vVb y fpr we know tbat tbe cube root of,
a product a A i9 fpund by multijilying tpgjetber tbe cube roots of.
tbe factors ( 1 56). - Hence^ also, ^we 9toid)e ^T by v^ ^^ 9^*
HentVfM
mbejt.
3 3 ,^
166. We further p^eive, Uiat 2 \/r is tqwA to vs a^ becaosa
2 is equivalent to VB ; that 3 v^a is equal to y/27 a» &nd b y^a is
equal to \/abhT, So^ reciprocallfy if' tbe nomber under the radi*
cal sign haaa Aictor which is a cnbe^ we may make it disappear
by placing its cube root 4f^are the sign^ For example^ instead
8 .9^ 3 __ . L 8
of V6ia ^e may write A\/ai and 5\/a instead of v^i25a«
8_ 8_
Hence vis is equal to. 2 v^ becauae 16 is equal Id S x S. .
167* When a number proposed is negative^ its cube root i»
not subject to the same difficulties that occurred in treating of
square roots. For, since the cubes of negative numbers are
negative, it foUows that the cube roots of negative numbers are
8_ 8
only negative. Thus, v— 8 is equal to — 2, and v^^r to ^^ 3.
3 3__ 8
It foUows also, that v"— 12 is the same as — vis, and that v— a
3
may be expressed by — \/a. Whence we see, that the sign — ,
when it is found after the sign of the cube root, might also hare
been placed before it We are not therefore here led to impos-
sible, or imaginary numbers, as we were in considering the
square roots of negative numbers.
CHAPTER XVh
t
Cf Powers in genefol*
168. Ths product^ which we obtain by midHplyifig a number
several Hmes by itsdf, is eatted apower. Thus, a square which
arises from the multiplication of a number by itself, and a cube
which we obtain by liultipljing a number twice by itself, are
powers. FFe say also in the former case, that the number is raised
to the secmtd degree, or to the second power ; and in the latter 9 that
the nufober is raised to the third degree, or to the third power.
169. We distinguish these powers from one another by the
number of times that the given number has been nscd as a (actor*
For example, a squaiip is called the second power, because a
I certain given number has bemi used twice as a factor ; and if
a number has been used thrice as a fs\ctor, we call the pro-
duct the third power, which therefore means the same as
tbeiettbe. > Multifly a niunber by itaelf tiU y4»uhave used it four
times as a fnotor, and yoii wHl^baye its fourth power, or what 19
commonly called the bi-!guadfcak» Frwi '. wliat has been paid it
will be easy to understand what is meant by the fiiltb, sixth,
seventh, &c«, power of a number. I only add, that the names of
these powers, after the fkirtk degree^ cease to have any other
bnt these numeral distinetiens.
170. To iUustrate this still furth*er, we may observe^ in the
first place, that the powers of 1 remain always the same; because,
whatever number of times we multiply 1 by itself, the produot
is found to he always 1. We shall therefore begin by repre-
aenting the powers of % and of 3. They succeed in thq following
<irdcr :
ri
48
JfjffDfVt
faebH.
Powers*
Of the namber 9A0t the number S.
Km
1 .A.
r ^ i
r If ^
L
s
s
II.
4
9
lU.
8
2r
IV.
16
81
V.
Sd<
843
VI.
64
729
VIL
128
8187
VIII.
256
6561
IX.
512
19683
X.
1924
59049
XI.
8048
177147
XIL
4096
531441
XIII.
8192
1594323
XIV.
16384
4782969
XV.
32768
14348907
XVI.
65536
43046721 >!
XVII.
181tor2
12914016S
XVHI.
868144
387480488 *
Bat the powers of the nmiber 1*0 are the nuirt remarictUe ;;
for on. these powers the system 0f out* trit&metic is founded; A
fcw of them arranged in order, and begimifaisf wttb th^ first
power, are as follbws : *
I. iL HI. n. y. VL
10. 100, 1000, 10008, 100000, 1000000, ftc
171. In order to lliustrate this atibject, and to consider it in a
tnore general mantiiBr, we may observe, tlii^the powers of any
number, a, succeed each other iil the Mlowini^ oi'der.
I. II. in. IV. ^"v. VI.
a, aUf aaOf aaaa, aaaadp aaaaamptx*
But we soon feel tlie inconvenience attending this mmner of
writing powers, which consists in the necelsity of repeating
the same letter veiy often, to axpress high powers ^ and the
reader also would hare no less trouble, if he were obliged to
count aH the letters, to know what power is intended to be
represented. The hundredth power, for example, could not be
conveniently written in this manner; and it would be still more
diflScult to read it. »
172. To avoid this inconvenience, a much more commodious
metliod of expressing such powers has been devised, which from
Ch«|i^ 16. Qf Bimfk ipiantiiies. 49
its extensive use deserves to be carefully explained ; vi^.. To
express^ for exaroplet the hundredth power, we simply wrfte the
number 100 above the number whose hundredth power we would
express, and a little towards the right-hand ; thus a^^^ means
a raised to 100, and repre»enU the hmidreih pewer ^ a. It must
be observed, that the name exponetU is given to the number writ-
ten abcroe that whose power, or degree, it represents, and which in
the present instance is 100.
17S» In the same manner, a* signifies a raised to 2, or tha
second power of a, which we represent sometimes also by aof
because both these expressions are written an4«imdersfood with
equal fiicility. But to express the cube, or the third power a a a,
wo write a^ according to the rule, that we may occupy less room. .
So a^ signifies the fourth, a' the fifth, and a^ the sixth power
of a.
174. In a word, all the powers of a will be represented by a,
a*, a*, a*, a*, a*, a'', a*, a*, a*®, &c. Whence we see that in
this manner we might very properly have written a^ instead
of a for the first term» to shew the order of the series dlore
clearly. In (act sl^ is no more tlutn a, as this unit shews that the
letter si i$ to be written ohly once* Such a. series of powers 10
called also a geometrical progression, because each term is
greater by one than the prodding.
175. As in this scries^ oF^^rs each term is found by multi-
plying the preceding^tern^ IjA^ which increases the exponent
by 1 ; so when any terak;js f^fik9^^rt may also find the preced-
ing one, if we divide by o^K^ause this diminishes the exponent
by U This shews that the term which precedes the first term a^
must necessarily be — , or 1 ; now. If we proceed according to
the exponents, we immediately conclude;^hat the term which
precedes the first must be a^. Hence we d/educe this remark-
able property ; that a^ u oonstanay equal to 1, however great or
small the value rf the number a may be, and even when a is noth*
iag; that is to saj, a^ is equal to 1.
176. We may continue our series of ^wers in a retrograde
order, and that 1x^^*0 different ways ; first, by dividing always
by Ti« an<l secondly^ diminishing the exponent by unity. And
Eul. Alg. 7
1 '
50
Jttgdntt*
Sect. 1*
it 18 erident Ibat* whether we follow the one or the other, the
terms are still perfectly eqaaL This decreasing series is
represented, in both formiSy in the following table, which must be
read backwards, or from right to left.
1
1
i
1
I
1
1
1
a
aaaaaa
aaaaa
aaaa
aua
ua
a
1.
1
1
1
0*
1
1
a*
1
2.
■
a-«
a-*
or*
or*
ar^
a*
17T. We are thus brought to understand the nature of powers*
whose exponents are negative, and are enabled to assign the
precise value of these powers. From what has been said^ it ap-
pears that,
«• "1 ri ; then
I- 1
»-«
H-a
> is equal to <
i
1 1
— ; or -a
-^m &C.
1
17B. It will be easy, fWrni the foregoing notation, to find the
ptywers qfaprod^, ab. Tliey mu$t evideMy he ab, or a» bS
a* b», a* b», a* b*, a* b», ^-c. Aid *A« powers effradioM vriU
befoand in the same manner ; for example those of g- are.
a« a* a» a* a» a« a^
»" a- a- a- •- •- •_ «
b»' b«' b»' M* £»" bP b^' •
179. Lastly, we hare to consider the powers of negatiyjj num-
bers. Suppose the given number to be — a; its powe» Mill
form the following series :
— a, +aa9 — aS +aS — o», +9^, tea
Chap. 17. qfaimfie qxianOitm. 51
We may observey that those powers only become negative
ivhose exponents are odd nnmbersy and tbat^ on the contrary^
all the powerSf which have an even number for the exponent »
are positive. So that the third, fiftbi seventh, ninth, &c.f pow-
ers have each the sign) — ; and the second, fourth, sixth, eighth,
&c. powers aro^ected witli the sign -{-.
CHAPTER xvm:
Cjftht calcuUUum of I^^s.
180/ We have nothing in particular to otiserve with regard
to the addition and subtraction of powers; for we only repre-
sent these operations by means of the signs + and — , when the
powers are different. For example, a' + a* ts the sum of tlx
second and third powers ^ a ; and a' — a^ is what remains
when we subtract the fourth power of sl from the fifth ; and neither
of these results can be abridged. When we have powers of the
same kind, or degree, it is evidently unnecessai'y to connect
them by signs ; a^ +a* makes 2 a', &c.
181. But, in the multiplication of powers, several things re-
quire attention.
First, when it is required to multiply any power of a by a,
we obtain the succeeding power, that is to say, the power whose
exponent is greater by one nnit. Thus a*, multiplied by a,
produces a^ ; and a', multiplied by a, produces a^. And, in
the same manner, when it is required to multiply by a the
powers of that number which have negative exponents* we must
add 1 to the exponent. Thus, a' ^ multiplied by a produces a^ or
1 ; which is made more evident by considering that a' ^ is equal
to — , and that the product of — by a being -, it is consequently
equal to 1. Likewise a*^ multiplied by a produces aj^, or
-^ j and a"^^, multipled by a, gives a"*, and so on. ^
182. Next, if it be required to multiply a power of ahj aa,
or the second power, I say that the exponent becomes greater
by 2. Thus, the product of a* by a* is a* ; that of n* by a* 5^
«2 Sgehra. Sect 1
a* ; tbat oFo^ by a^ ts a^ ; mA, mor6 geniBrallj, a» tmittiptMit
% a* mafrfs a*^'. WUh regard to nq^iive exponentSf we shall
liave a^9 or a, for the product cf vr^ hy a* ; for a^^ being equal
to — » it is the same as if we had divided aahj a; consequently
the product required is -*-, or a. So a-*, muUiplied iy a* pro-
duces a®, or 1 ^ and a^*, muUipKed by a», produces sr^.
183. It is no less evident thatt to multiply any power of a by
a^9 we must increase its exponent by three units ; and that
consequently the product of a** by a* is a^^. And whenever U
is required to multiply togetlier two powers of a, the product will Ite
also a power of ai and a power whose exponent will be tlit sum cf the
exponents of the two given powers. For example^ a^ multiplied by
a' will make a*» and a^ ' multiplied by a^ will produce a^ *, &c.
184. From these considerations we may easily determine the
highest powers. To find» for instance^ the twenty-fourth power
of 2^ I multiply the twelfth power by the twelfth power, because
£** is equal to 2*' x 2**. Now we have already seen that
2** is 4096 ; I say therefoi-e that the number 16777216, or the
product of 4096 by 4096, expresses the power required, 2*^.
185. Let us proceed to division. We shall remark in the
first place, that to divide a power of sl by sl, we mmt subtract 1
from the exponent f or diminish it by unity. Thus a', divided by
0, gives o* ; a®, or 1, divided by a. Is equal to a-* or — ; a**,
divided by a, gives o'"*.
186. ir we have to divide a given power of a by a*, we must
diminish the exponent by 2 ; and if by a^, we must subtract
three units from the exponent of the power proposed* So, in
generaU whatever power (f Ait is required to divide by another
power if ^9 the rale is always to ubtract the exponent of the second
from the exponent of the first of tliese powers. Thus «*', divided
by a', will give a» ; a« divided by o', will give o*"*; and a'^^f
divided by a*, will give ar^.
1 87. From what has been said above, it is easy to understand
the method of finding the powers of powers, this being done by
multiplication. When we seek, for example, the square, or the
nccond power of a^f we find a* ; and in the same manner we
Chap. 18. Of Simpie qumOUia. 53
find u^ > for the third power or the cube of aK To obtain the
square rfapotufr, we have only to douUc it» exponent ;for its cube,
we mu$t trifle the exponent ; and so on. The square of a" is
a** ; the cube of o^ is a*" ; the seventh power of a" is a''% &c.
188. The square of a*, or the square of the square of a,
being a^9 we see why the fourth power is called the bi-qnadrate.
The square of a^ is a* ; the sixth power has therefore received
the name of the sqnare^cubed.
Lastly, the cube of c^ being a*» we call the ninth power the
cubo-cube. No other denominations of tliis kind have been
introduced for powers^ and indeed the two last are very little
used.
CHAPTER Xmi.
Of Bo(ds with reUOioH to Powers in general.
189. SiircE the square root of a given number is a number^
whose square is equal to that given number ; and since the cube
root of a given number is a number* whose cube is equal to that
given number ; it follows that any number whatever being given,
we may always indicate such roots of it, that their fourth, or
their fifth, or any other^power, may be equal to the given num-
ber. To dtstingnisifk these di^rent kinds of roots better, we
sliall call the square rof t the second root ; and the cube root the
third root ; becpMfse, needing to this denomination, we may call
Aefoi&ih rootf that whose biqoadrate is equal to a given num-
ber ; and the fifth root, that whose fifth power is equal to a given
number, kc.
190. As the square, or s|^nd root, is marked by the sign
V/, and the cubic or tliird root^ the sign y', so the fourth root
is represented by the sign v 9 thi|i|ifth root by the sign ^ ; and
so on ; it is evident that according to this mode of expression,
the sign of the square rout ought td be ^Z. But as of all roots
this occurs most frequently, it has been agreed, for the sake of
brevity, to omit the number 2 in the sign of this root. So that
54
JIlgAra.
Sect 1.
when a radical sign has no number preftxed, this idways shews
that the square root is to be understood,
191. To explain this matter still furtlier» we shall here exhibit
the different roots of the number a, with their respective values ;
*_
<
8o thatconversely ;
The 2d
As theK
2dl
3d
4th
5th
Of
»root of >a.
6th^
a.
ja$ and so on*
TheSd
The 4th
The 5th
>power of <
4
B
Va J
Of
is equal to < a.
(It
\Jh
The 6thJ
and so on.
192. Whether the number a therefore be great or smalli ws^
know what value to affix to all these roots of different degrees*
It must be remarked also, that if we substitute unity for a» all
those roots remain constantly 1 ; because all the powers of 1
have unity for their value. If the number a be greater than l^
all its roots will also exceed unity. Lastly, if that number be
less than !» all its roots will also be less than unity.
193. When the number a is positive, we know from what was
before said of the square and cube f^ts, that ail the other roots
may also be determined, and will be real and possible numbers.
But if the number a is negative, its second, fourth, sixth, and
all the even roots, become impossible, or imaginary numbers ;
because ail the eroen powers, whether of positivef or of negative
number Sf are affected with the sign +• Whereas the ihirdf Jifth^
serenihf and all odd roots, become negativCf but rational; because
the odd powers of negative numbers, are also negative.
Chap. 19. ^ Qf SimpU quanmes. 53
194. We hare here abo an inexhaustible scoorce of new kinds
of surd, or irrational quantities $ for whenever the number a is
not actually such a power^ as some one of the foregoing indices
represents^ or seems to require* it is impossible to express
that root either in whole numbers or in fractions ; and conse«
quently it must be classed among the numbers which are called
irratioQid.
CHAPTER XIX.
Of the Method if representing Irrational JV\imAfrs by Fradumal
EoeponeniSm
195. We have. shewn in the preceding chapter, that the square
•f any power is found by doubling the ex|K>nent of that^wer,
i^id that in general the square, or the second power of a" » is
a'". The converse follows, namely, that the square root of the
power a*" is a" • and that it isfound by taking half the exponent of
^udpoweTf or dividing it by 2.
196. Thus the square root of a* is a^ ; that of a^ is a* ;
that of a* is a' ; and so on. And as this is general, the square
root of a^ must necessarily be a^ and that of a' a^. Con-
sequently we shall have in the same manner a' for the square
root of a^ I whence we see that a^ is equal to v^; and this
new method of representing the square root demands particular
attention.
197« We have also shewn that, to find the cube of a power as
a* , we must multiply its exponent by 3, and that consequently
tliecnbeisa'*.
So conversely, when it is required to find the third or cube
root of the power a'", we have only to divide the exponent by
S, and may with certainty conclude, that the root required is a".
Consequently aSor a, is the cube root of a^ ; a* Is that of a^ ;
a^ is that of a* ; and so on.
198. There is nothing to prevent us from applying tlie same
reasoning to those cases in which the exponent is not divisible
56 JBgAra. Sect* 1.
by 3, and concluding that the cube root of a* is a\ and that the
cttbe root of a^ is a#, or a ''• Conseiiiiently the thirds or
cube root of a also« or a ^ must be o^ • Whence it appears that
a^ is equal to \/tL.
199. It is the same with roots of a higher degree. The
fourth root of a will be a ^» which expression has the same valve
as v/a. The fifth root of a will be a^, which is consequently
equivalent to ^^ay and the same observation may be extended
to all roots of a higher degree.
£00. We might therefore entirely reject the radical signs at
present made use ofy and employ in their stead the fractional
exponents which we have explained ; howevei^t as we have been
long accustomed to those signSf and meet with them in all
books of algebra, it would be wrong to banish them entirely*
But tltere is sufficient reason also to employ, as is now frequently
done, the other method of notation, because it manifestly corres-
ponds with what is to be represented. In fact, we see iimiKMliate-
]y that a^ is the square root of a, because we know that the square
of a^t that is to say, a^ multiplied by a>, la equal to a ^ or a.
201. What has now been said is sufficient te shew how we
are to understand all other fractional exponents that may occur«
4
If we have, for example, a^, this means that we must first
take the fourth power of a, and then extract its cube or third
4 S
root ; so that a^' is the same as the common expression, va^^
s
To find the value of a^, ^e must first take the cube, or the
third power of a, which is a^f and then extract the fourth root
of that power; so that a^ is the same as y^as. Also a^ is
equal to v/o^f &c.
202, When tiie fraction which represents the exponent ex*
ceeds unity, we may express the value of the given quantity in
another way. Suppose it to be a^ ; this quantity is equivalent
to a ^, which is the prodnct of a* by a^. Now §ff being
Chlip.19. Of iMiiijite^iMm«/te#. ^^^ 5r
jr 10
equal to s/r, tt is evident that c» is equal* to a* v'tfT So a ^ ,
or a ^ is equal to a* Va ; and a ^.9 t^at is a *> expresses
'«^ Vo^* These examples are sufficient^ to iUustrate the great
utility of fractional exponents.
SOS. Their use extends also to fractional numbers : let there
besiven— =fWe know that this quantity is equal to -7; now
1
we have seen already that a firaction of the form -^ may be ex-
pressed by €r^ ; so instead of -*r=:: we may use the expression
9r\. In the satne manner^ -^ — is egvuZ to a'4. Again> let
the quantity be proposed ; let it be transformed into this,
A*' 8
-19 which is the product of a' by ar% ; now this product is equi-
valent to a^, or to a S or lastly to a Vo. Practice will ren-
der similar reductions easy.
204. We shall observe* in the last place, that each root may
be represented in a variety of ways. For y"^ being the same
as a^, and \ being transformable into all these fractions, |« ^, f,
tV tV^*> it is evident that v^a is equal to Vas, and to v^s and
to Va^« &nd SO on. In the same^ manner ^a, which is equal
to d^f will be equal toya»9 &nd to ^/as, and to x/a*. And
we see also, that the number a,pr a^, might be represented by
the following radical expressions :
\/a», \/a^f VflS Vfl^f occ.
205. This property is of great use in multiplication and
division : for if we have, for example, to multiply ^a by v/a«
we write va' for ^a, and v^as instead o! \/a ; in this man-
ner we obtain the same radical sign for both, and the multi-
plication being now performed, gives the product v^. The
same result is deduced from o*"^, the product of a' multi«
Eld. Mg. 8
58 J^ Mgehra. ' Sect 1.
plied by a^ ; for | + 1 iS j^, and consequently the product re^
quired is a^ or \/a'* T -
ft were required to divide Va, or a', by x/a^ or a', njf
^Ami&f Aar€ /or ^Af quciient 9? "^^ or 2J ^, that is say 9 a«
or x/Z
CHAPTER XX.
Of the different methods of calculation, and tf their mutual
connexion.
£06. Hitherto we have only explained the different metliods
of calculation : addition^ subtraction, m)iltiplication» and divis-
ion ; the involution of powei'Sy and the extraction of roots. It
H'lll not be improper therefore, in this place* to trace back the
origin of these different methods, and to explain the connexion
which subsists among them ; in order that we may satisfy our-
selves whether it be possible or not for other operations of the
same kind to exist. This inquiry will throw new light on the
subjects which we have considered.
In prosecuting this design, \\e shall make use of a new cha-
racter, which may be employed instead of the expression tliat
has been so often repeated, is equal to$ this sign is =, and is
read is equal to. Thus, when I write a = 6, this means that a is
equal to 6; so, for example 3^5 = 15. — X
207. The first mode of calculation, which presents itself to the
mind, is undoubtedly addition, by which we add two numbers
together and find their sum. Let a and b then be the two given
numbers, and let their sum be expressed by the letter c, we shall
have a -f 5 = c. So that when we know the two numbers a and
b, addition teaches us to find the number c.
208. Preserving this comparison a + & = c, let us reverse the
question by asking, how we are to find the number b, when ^e
know the numbers a and c.
It is required therefore -to know what number must be added
to a, in order that the sum may be the number c. Suppose, for
e?^amplei a=3 and c==8; so that we must have 3 + 6 =8;
Chap. SO. Of Simpk ((aantities. 59
I will evidently be foand by subtracting 3 from 8. So, in genera^
to find fry we must subtract a from c, whence arises 5 = c — a ;
for by adding a to both sides again, we have 6 + asc—- a + o^
that is to say = c, as we supposed.
Such then is the origin of subtraction.
fiOO. Subtraction therefore takes place, when we invert the
question which gives rise to addition. Now the number which
k is required to subtract may happen to be greater than that
from which it is to be subtracted ; as, for example, if it were
required to subtract 9 from 5 : this instance therefore furnishes
us with the idea of a new kind of numbers, which we call nega-
tive numbers, because 5 — 9 = — 4^
210. When several numbers are to be added together which
are all equal, their sum is found by multiplication, and is called
a product. Thus ab means the product arising from the multi-
plication of a by b, or from the addition of a number a to itself
h number of times. If we represent this product by the letter
c, we shall have abzzc; and multiplication teaches us how to
determine the number c, when the numbers a and b are known*
Sll. Let us now propose the following question : the numbers
a and c being known, to find the number 5. Suppose, for
example, = 3 and c = 15, so that 36=15, we ask by what
number 3 must be multiplied, in order that the product may bn
15 : for the question proposed is i*educed to this. Now this is
division: the number required is found by dividing 15 by 3;
and therefore, in general, the number ( is found by dividing c
I*
by a ; from which results the equation ft = — .
212. Now, as it frequently happens that the number c cannot
be really divided by the number a, while the letter ( must how-
e^er have a determinate value, another new kind of numbers
presents itself; these are fractions. For example, supposing
a = 4, c = 3, so that 4 5 = 3, it is evident that b cannot be an
integer, but a fraction, and that we shall have ft = |.
£13. We have seen that nHiltiplication arises from addi-
tion, that is to say, from the addition of several equal
quantities. If we now pi^oceed further, wo shall perceive
that from the multiplication of several equal quantities to-
60 JBgebru. Sect 1.
gether powers are derived. Those powers are represented in
a general manner by the expression a^, which signifies that the
number a must be multiplied as many times bjr itself^ as ia
denoted by the number b» And we know from what has been
already said, that in the present instance a is called the root^ b
the exponent^ and a^ the power.
S14. Further, if we represent this power also by the tetter e,
we have a* = c, an equation in which three letters a, ft, c, ara
found. Now we have shewn In treating of powers, how to find
the power itself, that is, the letter c, when a root a and its
exponent b are given. Suppose, for example, a^ 5^ and ft = 3,
so that c = 5 ' ; it is evident that we must take the third power
of 5, which is 125, and that thus c s 125.
SI 5. We have seen how to determine Hie power e, 1^ means
of the root a and the exponent ft ; but if we wish to reveiM the
question, we shaH find that this may be done in two ways, and
that there are two difierent cases to be considered : for if two
of these three numbers a, ft, c, were given» and k were reqatnid
to find the third, we should immediately perceive that this
question admits of three diSbrent soppositioaSf and consequently
three solutions. We have considered the case in whidi a and ft
were (lie numbers given, we may ttierefore suppose further that
c and a, or c and ft are known, and that it is required to dator^
mine the third letter. Let ns point out therefore^ before we
proceed any further, a very essential distinction between mro-
lution and the two operations whidi lead to it. When in
addition we reversed the question, it conld be done only in one
way ; it was a matter of indifference whether we took c and a,
or c and ft, for the given numbers, because we might indiffbr-
ently write a -f- ft, or ft -|- a. It was the same with multiplica-
tion ; we could at pleasure take the letters a and ft for each
other, the equation aft rrc being exactly -the same as bazzc»
Tn the calculation of powers, on the contrary^ the same thing
docs not take place, and we can by no means write ft" instead of
(f • A single example will be suflSicient to illustrate this : let
a=:5n and ft = S ; we have a* = 5* = 125. But 6« = S' = 243 :
two \ery different results.
SECTION II.
OF TB£ DIVTSRENT METHODS OF CALCULATION APPLIED TO
ooicvouND qvAvrrnibs*
y
CHAPTER I.
qf (Ae Addition of Ckmpound ^uantitiesp
ABTICUfi SI 6.
Wbbv t^i» Of note expressiong* coiuristing of several termsy
afe to be odde4 together, tbe operation is freqOeiitly represented
merely by signs, placing each expression between two paren-
theses, and connecting it with the test by means of the sign -f.
If it be required, for exanirie, to add the expressions a+b+c
and d + e+f, we rqHfsent the som thus :
(a+b + e) + (d+e+f).
£17. It is evident that this i|9 not to perform addition, bat only
to represent it. We see at the same time, however, that in
order to perform it actually, we have only to leave out the
parentheses ; for as the number d+e +fis to be added to the
other, we know that this is done by joining to it first -f d, then
+ ty and then +// which therefore gives the sum
The same method is to be observed, if any of the terms are
aflfected with the sign — > ; they must be joined in the same way,
by means of their proper sign*
£18, To make this more evident, we shall conrider an exam*
pie in pure numbers. It is proposed to add the expression
15 — 6 to 12 -^ 8. ir begin by adding 15, we shall have
Id -^ 8 -f 15 ; now this was adding too much, since we had only
to add 15 — 6, and it is evident that 6 is the number which we
have added too much. Let us therefore take thiaGawayby
writing it with the negative sign, and w^e shall have the true
sum, 12 — 8 + 15 — 6,
62 JUgdira. Sect.^
vhich shews that the sums art found by writing all the terms,
each xviih its proper sign,
219. If it were required therefore to add the expression
d — 6— -/ to a — h + c, we should express the sum thus :
a — b + c + d — e— /,
remarking however that it is of no consequence in what order
we write these terms. Their place may be changed at pleasure^
provided their signs be preserved. This sum mighty for exam*
ple^ be written thus : .
c — e + a — f+d'^b.
£20. It frequently happens, that the sums represented in
this manner may be considerably abridged, as when two or
more terms destit>y each other ; for example, if we find in the
same sum the terms -f a-^a, or da— i4a-f a: or when'two
or more terms may be reduced to one. Examples of this second
reduction:
Sa + ^a:=s5a; 7fr^-^Sb = + 46;
— -6c + 10c=: + 4c;
5a — 8a = — 3a; — 7 b + b=z^^6b;
— 3c — 4c= — 7c; •
2a — 5a + a:=z — 2a; — Sb — 5ft + 26= — 6b.
Whenever two or more terms, therefore, are entirely the same with
regard to Utters, their sum may be abridged : but tho^e cases
must not be confounded with such as these, 2aa-f 3 a, or
£ 5s .^ 54^ which admit of no abridgment.
221. Let us consider some more examples'of reduction ; the
following will lead us immediately to au important truth. Sup-
pose it were required to add together the expressions a + b and
a b; our rule gives a + b + a — b; now a + a=s2a and
5 ... b = ; the sum then is 2 a : consequently if we add together
the sum of two numbers (a +6) and their diflference (a •--6,)
we obtain the double of the greater of those two numbers.
Further examples :
3 a — 2 b — c
56 — 6c-f a
a^ — 2aab + 2abb
— aab + 2abb — 6*
4a+Sb — 7c a^ — 3aa6 + 4aft6 — 6*.
Chap. 2. OJ Compovmi ^mntiiie$» 65
CHAPTER II.
Ofih^ Subtraction of Compofwnd Quantities*
£!S2. If we wish merely to represent sabtraction» we inclose
each expression within two parenth^seg, connectingt by the sign
— ^9 the expression to be sabtracted with that from which it is to
be taken.
When we subtract, for example, the expression i^^e+f
from the expression a — i-fc^-we write the remainder thus :
(a_6 + c)— (d — e+/)|
and this method of representing it sufficiently shews, which of the
two expressions is to be subtracted from the other.
. 223. But if we wish to perform the subtraction, we must
observe, first, that when we subtract a positive quantity +&
from another quantity a, we obtain a— - b : and secondly, when
we subtract a negative quantity — h from a, we obtain a-f b;
because to free a person from a debt is the same as to give him
something.
224. Suppose, now, it were required to subtract the expres-
sion i — d from the expression a — c, we first take away ft;
which gives a-~ c — ^ fr. Now this is taking too much away by
the quantity d, since we had to subtract only h — d ; we must
therefore restore the value of d, and we shall thea have r
a^^c — ft + d;
whence it is evident, that the terms of the expression to be subtract-
ed must have their signs changed, and be joined, vnth the contrary
signs, to the terms of the other eocpression,
225. It is easy, therefore, by means of tliis rule, to perform
subtraction, since we have only to write the expression from
whicif we are to subtract, such as it is, and join the other to it
without any change beside that of the signs. Thus, in the first
example, where it was required to subtract the expression
d — e +/from a — ■ 6 + c, we obtain a — • 6 + c — d + c — f.
An example in numbers will render this still more clear. If
we subtract 6 >^ 2 4.4 from 9 — * 3 + 2, we evidently obtain
9 — 3 + 2 — 6 + 2 — 4;
for 9-— 3+2 = 8; also,6 — 2+4 = 8; now 8 — 8 = 0.
nH>^
64 «%<kf«^ Sect 9.
S2& Subtraetion being therefore subject to no dificolty, we
bave only to remark* that* ii there are found in the remainder
twot or more terms which are entirely similar with regard to
the letters* that remainder may he reduced to an abridged fonUf
by the same rules which we have given in additloB.
dsr. Suppose we have to mabtract from a + b* or frma the sua
of two quantities* their difference a -* A* we shall then have
a + b — a + b; now a-^a = 0*and 64. 6 = 26; the remainder
sought is therefore 2 6* that is to 8ay» the double of the less of
the two quantities.
228. The following examples will supply the place of further
illustrations.
aa+ab + bb
hb+ab^^aa
^aa.
3 a — 4b + 5dUi^ + Saab+Sabb+b^
^b^4c — 6a
9a — 66-f c
a* — Sanb^Sabb — ft«
6aa6-f-26^.
+ 5^*.
CHAPTER m.
Of the MuUipUoation of Compound ^uaniiUe$.
229« When it is only required to represent multiplication,
we put each of ttie expressions* that are to be multiplied together,
within two parentheses* and join them to each other* sometimes
without any sign* and sometimes placing the sign x between
them. For example* to represent the product of the two expres-
sions a — b+c and d — e +ff when multiplied together, we
write.
(a_j + c)x(d — e+/.)
This method of expressing products is much used* because it
immediately shews the factors of which they are composed.
230. But to shew how multiplicatton is to be actually per-
formed* we may remark* in the firat j^ace* 4hat in order to
multiply* for example* a quantity* such as a *^ ft 4. c* by 2* each
term of it is separately multiplied by that number ; so that the
product is
2a — 2i + 2c.
Chap* S» ^ Compcumd ^uantitiu. , 65
Now the same thing takes place with regard to ail other
numbers. If d were the nundier, by which it is required te
iiiulti{ily the same espression, we should obtain
ad-— bd-fcd.
£31. We supposed <I to be a positive number ; but if the factor
were a negative number^ as — « e, the rule heretofore given must
be applied ; namely^ that two contrary signs, multiplied together,
produce — y and that two like signs give -f.
We shall accordingty have
— ae + be — ce*
232. To shew how a quantity, J, is to be multiplied by a
compound quantity, d — e ; let us first consider an example in
common numbers, supposing that A is to be multiplied by 7 — 3«
Now it is evident, that we are here required to take the quad-
ruple of ttf ; for if we first take Ji seven times, it will then be
necessary to subtract 3 A from that product.
In general, therefore, if it be required to multiply by d— > e,
we multiply the quantity A first by d and then by e, and subtract
this last product from the first ; whence results dA — eA.
Suppose now ^9 = a i— ft, and that this is the quantity to be
multiplied by d — e ; we shall have
dA = ad — bd
eA^ae — b e
whence the product required = ad — bd — ae-^-be.
233. Since we know therefore tlie product (a — ^ b) x (<2 — ei)
and cannot doubt of its accuracy, we shall exhibit the same
example of multiplication under the following form :
a — b
d — c
ad — bd — ae + be.
This shews, that we must mtiSHply each term of the upper ex-
pression by each term of the loxver^ and that, with regard to the
signs, we must strictly observe the rule before given ; a rule
which this would completely confirm, if it admitted of the least
doubt.
EuUAlg. 9
« •
66 Jttgdn-^ Sect £.
254. It will bo easjy according to this role, to perfonii the
following example^ which is^ to multiply a + b by a — b:
a + b
fl — 6
aa + ab
—ab—bb
Product o a — 6 6.
S35. Now we may substitute, for a and b, any determinate
nuntbers ; so that the above example will furnish the following
theorem ; viz. The product of the sum of two numberSf multiplied
by their difference^ U equal to the difference of the squares of those
numbers, I'his theorem may be expi-essed thus :
(a + b)x(^a — b)=zaa — 6 6.
And fi*om this, . another theorem may be derived ; namely,
y.
The difference of two square numbers is always a product^ and
divisible both by Vie sum and by the difference ^ the roots of those
two squares.
236* Let us now perform some other examples : .
L)2a — 3
a+2
2aa^^Sa
+ 40 — 6
2 a a + a -^ 6.
IL)4aa — 6ii + 9
£a +S
8a* — 12«a + 18 a
+ \%aa — 18a + 2f
8 a» +27
V.
Cfaap. 8. (Jf Compound ^uanHties. ^7
UL) daa-^Sa6^6*
2a — 46
6a' — 4aa6*-^2ai6
— 12aab + 8abfr + 4(^
6a^ — 16 aab +6 ai( + 4 6'
IV.) aa+2a6 + 266
aa — 2ab + 2bb
a* + &a^b + 2aabb
— 2a'6-r.4aa66 — 4ah^
+ ^aabb +4ab^ +4b*
fl*+4 6*.
v.) 2aa — 3a6 — 466
Saa — 2a6 + 66
6o*— 9a'6 — I2aa66
-—4 a* 6 + 6aa66+8a6'
+ 2 aa 66 — 3a63 — 4 6*
6a^ — lda'6 — 4aa&6 + 5a6'--4 6«
VI.) aa + 66 + cc — a6 — ac — be
a+ 6 +c
a^+a66 + acc — aa6 — aac — a 6c
aa6+6' +6i:c^-ia66 — a6c — 66c
aac+66c + c^ — -.a6c — a co-Jicc
a» — 3a6c + 6' + c'.
23T. When tot have more than two qunntities to multiply to-
gethtTf it will easily be understood that, after liaving multiplied
ttoo of them together, we must then multiply that product by one
of those which remain, and so an. It is indifferent what order is
ehserved in these multiplications.
68 Mgehra. Sect Sk
Let it be proposed, for example, to ftnd the TftlaOf or product,
of the four following factors, vis6m
I. IL ' III. IV.
(a + b) (aa + ab + bb) (a— () (aa-^ab + bV).
We will first multiply the factors I. and II.
IL aa + ab + bb
I. a+ b
a* +aab + abb
+ aab + abb + b^
L II. a' +iaab + 2abb + b^.
Next let us multiply the factors III. and IT.
IV. ao — ab +bb
IlL a — b
a* — aab+abb
-^aab+abb — b^
III. IV. a» — 2aaft + 2a66~6»«
It remains now to multiply the first product I. II. by this
second product III. FV. :
a^ +Qaab + ^abb + b^ I. IL
a»— 2oa6 + 2o6 6 — »» III. IV.
, b* + 2 a' 6 +2 a* 6 6 + a» 6»
— 2a*6— 4a*66_4o*t*— 2aa6*
^a^bb + 4a^b* +4aab^ + S^ab'
_ a»6» — 2aa6^— 2oft«— 6«
And this is the product required.
238. Let us resume the same example, but change the order
of it, first multiplying the factors L and III. and then II. and
IV. Ibgether.
I. a+b
m. a—b
aa+ab
LIU.=:aa— »».
&hKf. 3. Of Compemd ((lumfiticl; ^
IL an^ah+bb
IV. aa—ab + bb
a^ +a*b + aabb
^^a* b-^aabb —.a **
aabb +ab* +b^.
n.IV.=a*+aaJ6 + 6*.
Then multiplying the two products L III. and IL IV.
ILl\.=za*+aabb + b^
I. llL=zaa—bb
a^ + a^bb + aab^
m
we have a'— 6* »
which 18 the product required.
239. We shall ptrform this calculation in a still different
manner, first multiplying the T*. factor by the IV*"". and next
the II''. by the III*.
IV. aa—ab+bb
I. a + b
a' — aab + abb
abb — abb + b^
I.IV.=^a»+6^
II. aa + ab + bb
III. a — ft
a^ +aab + abb «
^^aab — aftft— -ft'
iLin.=so»— ft*.
It Kinains to multiply the product I. lY. and 11. III.
I. IV. = c» + ft»
II.III.=o»— 6*
a* + a* ft*
^
and we still obtain a* — !•, as before.
7% J|r<!h«^ Sect 2.
240. It will be proper to illustrate this exAmjde by a numeri-
cal application. Let us make as 3 and ft =: s, we shall have
a + b=z5 and a — bz=zl; {tirther, aai=9f abzz6f bb=z4.
Therefore aa + a6 + 66= 19, and aa~ab + bb=z7. So that
the product required is that of 5 x 19 X I X 7, which is 665.
Now a* = 729f and b^ = 64, consequently the product re-
quired is a" -— (* = 665, as we have already seen.
CHAPTER IV.
Of the DivUon of Compmmi Quantities,
141. WhIIt we wish simply to represent division, we make
use or the usual mark of fractions, which is, to write the de-
nominator under the nnmerator, separating them by a line ; or
to inclose each quantity between a parenthesis, placing two points
between the divisor and dividend. If it were required, for
example to divide a -)- b by c -f d, we should represent the quo-
tient thus --j—f 9 according to the former method ; and thus,
(a + b): (c + d) accoi*ding to the latter. Each expression is
read a + b divided by c + </.
242. When \Jt is reqiiired to divide a compound quantity by a
simple ontf we divide each term separately. For example ;
6a — 86 + 4 c, divided by 2. gives 3 a — 4 64. 2 c ;
and(aa — 2a6) : (11) = 0—^*.
In the same manner
(a^ — 2aab + Saab): (a) = aa — ^ab+Sab;
(4aab — 6aac+^abc) : (2«) = 2a6 — Sac + 4lc;
(9 an be — \2 ab b c + 15 ab e c) : (S a b c) = S a — 46 +5 c, kc.
343. If it should happen tlutt a tena of tlie dividend is not
divisible by the divisor, the quotent is represented by a fraction,
\' b
as in the division of a + 6 by (i, whicK^ivcs 1 + — . Likewise,
(aa — a6 + 66) : faa) = l — 1— +^^
■i,., ^ a aa
For the same reason, iT <ve divide 2 a +6 by 2, we obtain
^ 4. ^ ; and iierc it may be remarked, tlmt we may w rite — 6,
Chap. 4. Of Compound ^vkintities. 71
hi h
instead of -r, because ^ times 6 is equal to — • In the same
manner — is the same as - b, and — the same as -- 6« &c.
S 3 3 S
244. But when tlie divisor is itself* a compound quantity,
division becomes more difficult. Sometimes it occurs wiiere we
least expect it ; but when it cannot be performed, we must con-
tent ourselves with representing the quotient by a fraction, in
the manner that we have already described. Let us begin by
considering some cases, in which actual division succeeds.
245. Suppose it were retjuired to divide the dividend ae — be
bythedivMor a-— ft, ffte quotient must then be such as, tttien
multiplied by the £visor a •— b, will produce tlie dividend a c — be*
Now )t is evident, 'that this quotient must include c, since with*
out it we could not obtain ac. In order, therefore, to try
whether c is the whple quotient, we have only to multiply it by
the divisor, and see if that multiplication produces the whole
dividend, or only part «f it. In the pi*esent case, if we multiply
a— -fr by e, we have a c — be, which is exactly the ^fividend^
80 that c is the whole quotient. It is no less evidentt that
(aa + ab^i (a + b)=za; (Saa — 2a6) : (So — 2 6) = fl;
(6 fl a — 9ab): (£ a — 3 ft) = 3 a, &c,
246. We cannot Jailf in this way 9 to find a part of the quotient;
}f, therefore^ what we have foumlf when multiplied by the dhnsor^
does not yet exhaust the dividend^ we home only to divide the
remainder again by the divisorf m order to obtain a second part of
the quotient; and to continue the same methodrUntU we have found
the whole quotient.
Let us,, as an example, divide a a -f 3 a k+ S ft 6 by a -f ft ; it is
evident, in the first place, tliat the quotient will include the term
# II, since otherwise we should not obtain a a. Now, from the
multiplication of the divisor a + ft by a, arises aa + ab; which
quantity being subtracted from the dividend, leaves a remainder
Sa ft + 2 ft ft. This remainder must also be divided by a -f ft ; and
it' is evident that the quotient of this division must contain the
term 2 ft. Now £ ft, multiplied by a + ft, produces exactly 2'a ft +
2 ft ft ; consequently a -f 2 6 is the quotient required ; wliicli^ mul-
72 JKgOra. Sects.
tiplied by the divisor a + ft, ought to produce the dividend
«, aa+Sab + fibb. See the whole openition :
a + b)aa + Sab+f^bb(a+ftb
aa + ab
ftah+2bb
2ab + 2bb
0.
247. This operation wUl be JadUialed if we choou one oj the
terms tf the divisor to be written Jirstf and then, in arranging the
terms tf the diviiendf begin with the highest powers of that first
term of the dhrisor. This term in the preceding example was a ;
the following examples wiU render the operation more clear.
a~ b) fts ..Saab + da 6 6—6' (aa— 3ab + (b
a^,mm,aab
2aab + Sabb
Uaab + 2abb
— ii^^..— — —
a66 — 6'
abb — b^
0.
^> 11 hi
a+b^ao'-^bbifl^^b
aa + ab
— ab^bb
^ab—bb
0.
Sa— 26)18aa^8 56(6a + 46
18aa— 12ab
IS ab — 8bb
IZab-^Sbb
0.
i
Cb^4. (^ C(mpQmd (fwniMies. 73
a' +aab
-^aab 'i'b*
— aab — abb
abb + b^
abb + b*
0.
2a — b) 8a*— 6* (4aa + 2ab + bb
8a* — 4ao6
4aa6— 6*
4aafr-— 2abb
■^1
2 a 6 & — 6»
0.
aa-^2a6+^^) A^ — ^^' b + 6 a abb — 4ab* +b^
aa — Zab+bb)a^ — «a* b-^aabb
.— 2a*6 + 5aa66 — 4a 6*
— Sa'i + 4aa6& — 2ab*
<
aabb^-^^ab^ +6*
a abb — 2 a 6'* +6*
II I p » ■!■■ ■
.0.
-^—
aa — £afr+466)a^ +4aa6(+ I6^\aa+2a&^466
a* — 2fl* fr + 4aa66
2a» ft + 166*
4aa66— 8afr> + i66«
4aabi — 8a6^ + 166«
0:
JSuI. JUg. 10
/
«
74 Mgdnu. Sect £.
a^ — 2a* i + 2aaft6
Zaahb—Aah^+Ah^
Staab b — 4 a 6* + 4 M
0.
1 — Sx+Sxx^^x^) I'^ZX + XX
-^Sx + dxx — lOar'
Sxx — 7a* -f 5ar*
Sarx— 6a;* +3a:'*
. — a:* +2a:*-r^'
— a?* +Sa:* — ar'
0.
CHAPTER V.
Of the Besdiitm ef Fradums Mo it^Me series.
248. Whei^ the dividend is not divisible by the divisor, fh»
quotient is espressedf as we have already observed, by a frac-
tion.
Thus, if we have to divide 1 by 1 — a, we obtain the fraction
— — • This, however, does not (mvent us from attempting the
«
division, according to the rules that have been given, and con-
tinuing it as far as we pieaae. Vfe shall not M to find the true
quotient, though under different forms.
Chap. 5. Of Compound ^anttKes* 75-
£49. To prove this, let us actually divide the dividend 1 by
the divisor 1 — a, thus :
1— a)l(l + .-^j or, 1 — 0)1(1 +a+-il
1 — a 1 — a .
remainder a "o '
a — aa
remainder a a
To And a greater number of forms, we have only to continue
dividing a a by 1 — a ;
1 — a) aa («« + , --> then 1 — a)a^ (a« +-
aa — tt* a^ — o*
and again 1 — d)a^ {a^ -{- ^ZI
a
a* — a'
a'f &c.
250. This shews that the fraction -— — may be exhibited un-*
i —a
der all the following forms :
III.) 1 + + 00+7^5 IV.)l+o + oo + a« + --2-.}
^ Y.)l+a + aa+a^+a^+ ^5_,&c.
Now, by considering the first of these expressions, which is
a 1 ■- n
1 4. , and remembering that 1 is the same as : , we
1— a 1 — a
have
l + i ! = ■; — ! + •: =-^ — -^ =
I— fl 1— a^l— a 1— fl 1 — a*
If we follow the same process with regard to the second
expression 1 + a + , that is to say, if we reduce the Ib*
1 "^O
76 JUgebra. Sect 2.
tegral part 1 + a to the same denominator 1 -— o^ we shall have
' ~^^ f to which if we add + r-— » w® s^JiaU **»'« — a ^ + " ?
1 — a 1 — a 1 — o
1
that is to say, .
In the third expression, X+a + aa + - — —f the integers
1 — a*
reduced to the denominator 1*^ a make : and if we
1 — a '
a^ 1
add to that the fraction >, we have<: ; wherefore all
these expressions are equal in value to ^^ , the proposed
fraction.
251. This being the case, we may continue the series as far
as we please, without being under Iho necessity of performing
any more calculations. We shall therefore have
1 . ^ a
s
= 1+a + aa + a^+a^ +a' ^a^ ^u'' ^
1 — a • 1 •— a
or we might continue this further, and still go on without end.
For this reason, it may be said, that the proposed fraction has
been resolved into an infinite series, which is
to infinity. And there are sufficient grounds to main tain, 4hat the
value of this infinite series is the same aa that of the friiction
I
■ •
I— a X
£52. What we have said may, at first, * appear surprising ;
but the consideration of some jiarticular cases will make it easily
understood.
Let us suppose, in the first place, asl ; our sen k€' will
become 1 + 1 + 1+ i + i + i+ 1, &cw^ The fraction , ,
1 —-11
to which it must be equal, becomes -r. Now^ we btfore remai*k-
cd, that -7; is a number infinitely great ; which is^ thereihre^
here confirmed in a satisfactory manner.
Chap. 5. Cf Compound {^nt%He$, 77 ^
But if we suppose a=s 2, our series become s: 1 + 2 +4 4- 8
+ l6 + S§^ + 64f &c« to infinity} and its value must be r—^
tbat is to saj, — — = — 1 ^ which at first sight w ill appear ab-
surd. But it must be remarked, that if we wish to stop at any
term of the above series, we cannot do so without joining the
fraction which remains. Suppose, for example, we were to stop
at 64, after having written 1 4.24.4 + 8 + 16 + 32 + 64, we
138 \2S
must join the fraction ■: :, or — -, or — 128 : we shall
therefore have 127 — 128, that is in fact — 1.
Were we to continue the series without intermission, the frac-
tion indeed would be no longer considered, but then the series
would still go on.
25S. These are the considerations which are necessary, when
we assume for a numbers greater than unity. But if we suppose
a less than 1, the whole becomes more intelligible*
For example, let a = ^ ; we shall have
1 _ 1 _ 1_
which will be equal to the following series :
1 +i +i + | +7V +7T + A +TTt» *c. 4o infinity.
Now, if we take only two terms of this series, we have 1 +|y
and it wants |, that it may be equal to = 2. If we take
three terms, it wants ^ ; for the sum is 1|. If we take four
terms we have 1^, and the deficiency is only |. We see, there-
fore, that the more terms we take, the less the difierence becomes,
and that, consequently, if we continue on to infinity, there will
be no diflPerence at all between the sum of the series and 2,
the value of the fraction .
1 —a
254. Let a=4 ; our fraction will be = 7 r = i =14,
' 1 —a l""y
which, reduced to an infinite series, becomes
and to which ^ ^^ ■ is consequently equal.
d
78 Jlgdn-a. Sect £•
%
' When we take two terms^ we have l-^t and there wants |. It
we take three terms, we have i^, and there will still be wanting
iV* Take four terms, we shaH have l|^, and the difference ifi
7^. Since the error, therefore, always becomes three timet
less, it must evidently vanish at last
255. Suppose a = 4 : we shall have =: =- = 3, and
* I— a I— "I
the series 14.|4.|4.^+|«+ tA* &c. to infinity. Taking
first i^y the error is 1^ ; taking three terms,, which make i^9
the error is | ; taking four terms we have 211, and the error
M IS
IS jj.
256. If a = i, the fraction is ; ; = — = U ; and the se-
* 1 -^J i ^
ries becomj^ 1 + ^ + tV + t^ + ttt» ^^ "^^^ ^^^ ^^^ terms
making 1 +^, wUl give ^ for the error ; and takiog one term
more, we have l-/j, that is to say, only an error of ^V*
257. In the same manner, we may resolve the fraction _ . ,
into an infinite series by actually dividing the uuQievator 1 bj
the denominator 1 + a, as follows :
i+a)l(l — a + aa — a«+o*
1+a
a
aa
aa + a^
fl»
r4
a'
Whence it follows, that the fraction -r— is equal, to the series,
l^a + aa~a'+a^-— a' +o«— a^,&c.
Chap. Si . - Of Compomid ifuanhUes. f 9
258. If we make a sly we have this remarkable comparison :
* =i = l — 1 + 1 — 1 + 1 — 1 + 1 — 1, &c. to infinity.
1 + fl
This will appear rather contradictory ; for if we stop at — U
the series gives ; and if we finish by + 1, it gives 1. But this
is precisely what solves the difficulty ; for since we must go on
to infinity without stopping either at — 1, or at + 1, it is evi-
dent that the sum can neither be nor 1, but that this result
muHt lie between these two^ and therefore be = |.
£59. Let us now make a = ^^ and our fraction will be
— --.j^ = |, which must therefore eipress the value of the series
l—i + i — i.+ TV — VT+A'&c. to infinity.
If we take only the two leading terms of this series, we have ^f
which is too small by |. If we take three terms, we have |,
which is too much by -^-g. If we take four terms, we have f
which is too small by ^^^ &c.
£60. Suppose again a ^^ ; our fraction will be = ■ ^ = |,
and to^his the series 1 — -J + 1 — iV + tt — ttt + rjj9 &c.
continued to infinity, must be eqiiah Now, by considering only
two terms, we have |, which is too small by i^. Three terms
make ^, which is too much by ^. Four terms make ||, which
is too small by ^|^'ihd so on.
£61. The fraction — ; — may also be resolved into an infinite
1 + a "^
aeries another way ; namely^ by dividing 1 by a + 1, as follows :
m
1
a
a aa
1
aa
1 1
— 4. —
«a ^a*
a>
a* a*
1_
a*
ConsequeQfljrf our fraction , » is eqoal to the infinite
senes -r- + -i^^i + -r — ri» «c. Let us make a = 1^
and we shall have the series
— 1 + 1 — 1 + 1 — lf&c=:|, as before*
And if we suppose a = £9 we shall have the aeries
£62. In the same manner, by resolving the general fraction
— >-T into an infinite series, we shall have,
a +
\
\
Chap. 5/ Of Campaund qmnlitUs. ^r
»^ , e he hbe b^ e
a + b) c i + — I -J-
»c
\
c +
a
6c
""""
a
be
bbc
a
aa
bbc
aa
bbc 6' o
aa a*
6»c
a»
6» c M c
,^ a* a'*
6*c
Whence it appears, that we may compare ^ with the series
c 6c 66c 6^Cg A*fi-A_
— — . — f- — -, &c. to infinity.
a aa a^ a*
Let a = 2, 6 = 4^ c ss 3, and we shall have
— 1-r = -i— = f = 4 = I — 3 + 6 — 1 2, &c.
fl -f. 6 2 + 4 T f f T f
Let a := 10, 6 2= 1, and c = 11, and we have
^ _ ^1 1 11^^ 111 11 _ 11 JKrr
J-j7J — jQ I 1 — 1 — TT "- ITT +TTTT "^TTTTT* «**^*
If we cQBsider only one term of this series, we hate 4t« which
is too much by yV * ^^ ^^ ^^® ^^^ terms, we have ^yt^^ which
is too small by ^^^ ; if we take three terms, we have 4ttt»
which is too much by ttVt« ^^«
263. When there are more than two terms in the divisor, we
may also continue the division to infinity in the same manner.
Thus, if the fraction ■ were proposed, the infinite
series, to which it is equal, would be found as follows :
Eul. Mg. 11
/
/
a— -aa
oA
— a* +<*'*"- fl^
a«
a« — .flT +a'
a*
We have therefore the equation of
1 *
:: ; =:l+a— «* — a* + »*+a^— a* — a*% &c.
1 —a + a a
Here^ if we make a = 1^ we ^ave
1 = I + I — 1 — 1 + 1 + 1 — 1 — 1 + 1 + 1, &c.
vhich series contains twice the series found above
1 — l + l — 1 + 1, &c.
Now^as we have found this=:|,it is not astonishing that we should
find !• or 1 9 for the vakie of that which we have just determined.
Malie a = ^^ and we shall then have the equation
t
Suppose a = |» we shall have the equation
-7— y— l-h-f — ^T — ¥T + TITT* ***^»
If we take the four leading terms of Uiis series^ we have ^^^,
ifrbich is only ^l^* less than f •
Suppose again a = |, we shall have
2. 9 1j_t 8 _16i 64 gfp
7— T— *+y — JT — TT + T¥T» *^^*
This series must therefore be equal to the preceding one ; and
subtracting one from the other, ^ — ^^ — tt + Vt^ ™"st be 5= 0.
These four terms added together make — -/f
Caiap, €. Of Campaumd ^[itantUies. ts
364. Tlie o^tb^dy which we haveexplainedf serves te resotre^
generally, all fractions into infinite series ; amU therefore* it is
often fottiui to be of t4ie greatest ntiUty. Furtlier, it is renark-
able# that an wJMie <ertes, though U never ceases, inay have a de«
ierminate valuta It may be added, tliat from this branch of
mathematics inyentions of the utmost importance have been de-
riTed, on which account the subject deserves to be stodied with
the greatest attention.
CHAPTER VI.
Of the Squares of Compound ^uauHHes.
265. VTheit it is required to find the square of a compound
quantity* we have only to multiply it by itself and the product
will be the square required.
For example, the square of a -f A is found in the following
manner :
a + b ^
11 + 6 ^
aa + ab
&b+bb
aa + Qab + bb»
266. So that, when the root consists of two terms added together,
as a +if the square comprehendSf Ist, the square of each term,
namely, a a and bh; 2dly, twice the product of the two terms^ name-
ly, 2 a ft. So that the sum aa + 2ab + bb is the square of a +b»
Let, for example, a = 10 and 6 = 3, that is to say, let it be requir-
ed to find the square.of IS, we shall have 100 +60 +9, or 169.
267. We may easily find, by means of this formula, the
squares of numbers, however great, if we divide them into two
parts. To find, for example, the square of 57, we consider t{(at
this number is = 50 + 7 ; whence we conclude that its square is
= 2500 + 700 + 49 = S249.
268. Hence it is evident, that the square of a + 1 will be
aa+2a + l: now sincethe square ofa is a o, we find the square
84 Jl^Ara. S^£«^
a+ 1 by aflding to that S a + 1 ; and it must be observedy tbat
this £ a + 1 is the sum of the two roots o and a 4> 1.
Thus, as the square of 10 is 100, that of 1 1 will be 100 -|- 21.
The square of 57 being 3249, that of 58 is 3S49 + 115 = SS64.
The square of 59 = 3364 + 117 = 3481 ; the square of
60 = 3481 + 119 = 3600, &c.
S69* The square of a compound quantity, as a +&> is repre-
sented in this manner : (a + 6)'. \/% have then
(a + 6)* = fl a + 2 a * + J fc,
whence we deduce the following equations :
(a + 1)* =aa + 2a + l; (a + 2)* =aa + 4a + 4 ;
(a + S)» =aa + 6o + 9; (a + 4)* =aa + 8a + 16 j &c.
270* If the root is a — b, tlu square ^itisaa— 2ab + bb,
ivliich contains also the squares of the two temiSf bu^ in such a
manner that we must take from their sum twice the froinct of those
two terms.
Let, fur example, a » 10 and 6= — 1, the square of 9 will be
fiiund = 7 00 — 20 + 1 = 81.
271. Since we have theequation (a— » 6)'= aa — Zab + hb^ we
shall have (a — l)'=aa — 2a + l. Thesqnareqfs, — lisfoundt
therefore, by subtracting from tk^the sum of the two roots a and
a — 1, namelf/fQsL — 1. Let» for example, a = 50, we hi^e
aa= 2500, and a — 1 = 49 : then 49* = 2500 — 99 = 2401.
272. What we have said may be also confirmed and illustrated
by fractions. For if we take as the root 1+4 (which make 1)
the squares will be :
Further, the square of ^ — ^ (or of |) will be
1 1 J- 1 — J
t — T + T — JT*
273. n hen the root consists of a greater number of (ermsj
the method of determining the square is the same. Let us find,
for example, the square of^ + b + c.
a + b + c
a + b + c
na+ab + ac +bc
ab + ac+bb + bc + cc
aa+2afr + 2ac + 66 + 26c + cc.
Ch9^ ^ tiff Compound (Quantities. 85
We see that it indudeSf jirsU the square of each term of the rootf
and beside that, the douUe>products of tiiose terms muUipUed two by
two,
274. To illustrate this bj an example, let us divide the num-
ber 256 into three parts, 200 + 50 +£ ; its square will then be
composed of the following parts :
40000 256
2500 •• 256
36
20000 15.^6
2400 1280
600 512
65436 65536
which is evidently equal to the product of 256 x 256.
275. When some terms of the root are negative, the square is
still found by the same rule ; but we must take care what signs we
prefix to live double products. Thus, the square of a — 5 — c be*
ing aa + bb'\'Cc — 2a6 — 2ac+25c, if we represent the
number 256 by 300 — 40 •— 4, we shall have.
Positive Parts. Negative Parts.
+ 90000 ~ 24000
1600 -^ £400
320 -,
16 —26400
+ 91936
— 26400
65536^ the square of 256^ as before.
CHAPTER Vn.
Of tht Extractum of EooU applied to Compound Quantities.
Sr6. Ik order to gire a certain rule for this operation* we
niiist consider attentively the sqoare of the root a + h, which is
aa + ^ab + bb, that we may reciprocally find the root of a
given square.
277. We must consider therefore^ first, that as the square
aa + Zab + bb\B composed of se vei*al terms^ it Is certain that the
root also will comprise more than one term ; and that if we
write the square, in such a manner that the powers of one of the
letters, as a, may go on continually diminishing, the first terra
will be the square of the first term of the roof. And since, in
the present case, the first term of the square is a a, it is certain
that the first term of the root is a.
278. Having, therefore, found the first term of the root, that
is to say a, we must consider the rest of the square, namely,
2 a 6 + ^ 6, to see if we can derive from it the second part of the
root, which is ft. Now this remainder 2ab + bb may be repre-
sented by the product, (2 a + 6) 6. Wherefore the remainder
having two factors, 2 a + 6 and 6, it is evident that we shall find
the latter, h which is the second part of the root, by dividing
the remainder 2 a b -|. 6 6 by 2 a + 6.
279. So that the quotient, arising from the division of the
above remainder by 2 a + b, is the second term of the root re-
quired. Now, in this division we observe, that 2 n is the double
of the first term a, which is already determined. So that
although the second term is yet unknown, and it is necessary,
for the present, to leave its place empty, i\'e may nevertheless
attempt the division, since in it we attend only to the first term
2 a* But as soon as the quotient is found, which is here bf we
must put it in the empty place, and ^hus render the division
complete.
280. The calculation, therefore, by which we find the root of
the square aa + Qab + bb, may be represented thus :
i
(j^fk f • ^ Campmni ^nKtie^. 78
aa
2a&+fti
0.
£81. We BMj, in the same manner, find tlie square root of
other compound quantities, provided they are squares^ as the
following examples wiH shew.
aa
2a + 3ft)6ai+9fr6
^ah + ^hh
0.
4aa— 4aft + fti(2a — i
Aaa
— 4ab + bb
X
0.
9pp + Z4pq + 16qq{Sp+4q
9pp
^p+4q)94pq + ieqq
^pq + l6qq
0.
25a?af— 60a: + 36(3a: — 6
25 XX
lOar — 6) — 60a: + S6 .
— 60 a: + 36
0.
88 Mgdfm. Sect 2.
28^. When there is a i^mainder after the dirision, it is a
proof that the root is composed of more than two^ terms. We
then consider the two termd already found as forming the first
part» and endeavour to derive the other from the remainder, in
the same manner as we found the second term of the root The
following examples will render this operation more clear.
aa + lah — Zae — 2b£4-Aft4-cc(a-f.i— ^a
aa
2a-f-2fr — c) — 2ac^*2frc-f cc
— 2ac — 2frc-f cc
0.
a'* + 2a*-f3aa + 2a + l(aa + a-fl
Zaa+ay Sa^ +3aa
Za*+ a a
2aa+2a + l
0.
a*
Zaa — 2a6) — Aa^h + %ah^ +Ah^
— 4a^ h^Aaahh
Zaa — Aah — 266) — 4aa66 + 8a6' 4-4 6*
— 4aa66 + 8a6'+4 6«
0,
Chap. 7. Of Campcmd quantiiks. 89
a«*-6a^fr+15a^ft6~20a'b* + 15aaft*— 6a5'+6«
a* (a^^^S^ab + Sahb — h^
2a*— 3aa6) — 6 a' b + 15 a« b ft
— 6a» 6+ 9a*ft6
2a'— 6aa6 + 3aft6)6a^ftb — 20a3 6' +15aa6^
6fl*66 — 18a»6' + 9aa6*
2a3— 6aaft + 6ai6~fr3) — 2a>6' + 6aaft«— 6ai' 4-A«
— 2a*6«+6aaM — 6afr'4.i*
0.
283. We easily deduce from the rule which we have explain-
edy the method which is taught in books of arithmetic for the
extraction of the square root. Some examples in numbers :
• »
529 (23
4
1764
16
(42 2904 (48
16
43) 129
129
82) 164
164
88) 704
704
•
0.
0.
0.
• •
4096 (64
36
9604 (98
81
124) 496
496
188) 1504
1504
0.
0.
15625 (125
1
998001 (999
81
22) 56
44
189) 1880
1701
245) 1225
1225
1989) 17901
17901
0.
0,
EuLMg.
m
90 MgAriu Sect. 2.
£84. But when there is a reaainder after the whole operationf
it is a proof that the number proposed is not a square^ and con-
sequently that its root cannot be assigned. In such cases^ the
radical sign, which we before einployed» is made use of. It is
written before the quantity^ and the quantity itself is placed
between parentheses, or under & line. Thus, the square root of
aa + hh\s represented by %/{aa^b\ov by Va a^h b ; and VCI— xx),
or v'i— jp^i expresses the square root of 1 — xx. Instead of
thai mdical Bfuo^ we nay use the fractional exponent ^ and
represent the square root x){aa + bh, for instance, by (a a + 6 h)^f
or by 4i«+6 6") ^*
CHAPTER VIII.
^ Of the calctdation of Irrational ^uafUitu$»
285. >Yhen it is required to add together /two or more irra-
tional quantities, this is done, according to the method before
laid down, by writing all the terms in succession, each with its
proper sign. And with regard to abbreviation, we must remark
that initead of v*" + V^T ^ e x a mp le, we write 2 ^/iTi and that
^^ — ^^ =r 0, because these two terms destroy one another.
Thus, the quantitUs S + ^/T and 1 + 4^2^ aided together, make
4 + 2 \/T or 4 + V8; the sum pf 5 + v'J'and 4 — \/2' is 9 ;
and that of 9 vs" + ^ Vi" ^^^ V3" — \/^ is 3 \/T + 2 v^E
286. Subtraction also is very easy, since we have only to add
tlie proposed mrariiers, changing first their signs : the following
example will shew this : let us subtract the lower number from
the upper.
4 — V2"+2vr— 3V5" + 4\/6"
i+2vr— sva"— 5vr+6v^6"
5 — 3 vF + 4 x/T + 2 s/T— 2 V6"
28T. In multiplication we must recollect that vaT tnuUiplied
hy Va produces a ; and that if the numbers which foUaw Vie sign
^ are different, as a and b, we have Vab for the product rf \/T
multiplied by vbT After this it will be easy td'perform the fol-
lowing examples :
Chap* 4. ^ (if Compound (^luuttities. 91
1+V2" 4 + 2^2]
+ V2"+2 — 4v'r— 4
I •
1 + 2^8" + 2 = 3 + 2^3" 8 — 4=4
288. What we have said applies also to iiMigiiwrj iioantilies ;
we shall only observe farther, that v^^ nmUM^ied by V^TT jiro-
duces — a. ^
If it were required to find- the cube of — 1 + V^^ we
should take the square of that number, and then multiply that
square by the same number : see the operation :
1—2^1:7 — 8 SB — 2— gv^HJ
— 1+ v'^^^
^
2+2
2 + 6 = 8.
289. In the division of surdi, we harve oidy to express the pro-
posed quantUies in the farm (f a fraction ; this may he (hen chang-
ed into another expression having a rational denominahfr. For if
the denominator be a + v^ for example^ and we multiply both
it and the numerator by 11 — y? the new denominator will be
a a i— b, in which there is no radical sign. Let it be proposed
to divide S + 2 vF by 1 + vsl we shall irst have f + ^ ^^ -
Multiplying now the two terms of the fraction by I — -v/37 we
shall have for the numerator :
9i MgAnu Sect 2.
1 — va"
5 + 2^2
— svr — 4
and for the denominator :
1 — v/a
1 +\/2_
1 — S= — 1
Our flew fraction therefore is — ~-- 1 and if we again
multiply the two terms by — 1, we shall have for the numerator
VF + h <^d for the denominator -f- 1. Now it is easy to shew
1. — S + 2 ♦/§"
that V2 + 1 is equal to the proposed fraction j— — -= ; for
y/T + 1 being multiplied by the divisor 1 -f VC ^^9
+ V2 +2
wO have 1 +2^2 +2=3 + 2^2.
Another ei sample: 8 — Sy/T divided by 3 — Sty/T makes
- — ■——. A* ^wltiplying the two terms of this fraction by
3 — 2 v^2
^ + 2 v^ we hav. ^ f<>^ ^I'^ numerator^
8 — 5^2"
S +2V2"
^ I6V2" — 20
24 + v '2 —20 = 4+^51
^ and for the denominator^
Chap. 8» Of Compound ^fuintities. 99
3 — fivT
3 + 2 v/a"
9 — 6^/2
+ 6 V2" — 8
9 — 8 = + 1.
Consequently the quotient mil be 4 + \/2. The ^th of this
may be proved in the following manner :
4+ v£
3 — 2v/2
is+sv/r
— 8^/2"— 4
12—5^2 — 4 = 8—5^^2"
290. In the same manner, we may transform such fractions
into others, thai have rational denominators. If we have, for
example, the fraction - — - — ;=-, and multiply its numerator and
denominator by 5 -f 2 v^ ^^ transform it into this
5 + 2v/6
1
= 5+2^6.
2
In like manner, the fraction — assumes this form,
2 + 2 y/Z T __ 1 + v^ZT
— 4 —2
And i^±^become8 = li±i^=ll+2V30.
291. FF%«n fAe denominator contains several termSf we may in
the same manner make the radical signs in it vanish one by one^
Let the fraction --= -= 7= be proposed : we first mul-
tiply these terms by \/Td + \/T + V^T ^^^ obtain the fraction
J* " — -=——. Then multiplying its numerator and denom-
5 — 2 ^^6 1^0
inator by 5 + 2 v^T ^^ have 5 yio + 11 ^^2"+ 9 \/z + 2 ^60.
94 JOgAm. Sect. d.
CHAPTER IX.
Of Cubes, and the Eoctraction rf Cvibe Roots.
292. To find the cube rf a root a + b, we only multiply its
squareaa + 2ab + bfragain by a + b, tlius^
na + Zab+bb
a + b
a^ +Zaab + abb
aab'\-2abb + b^
and the cube will be =a* +daa&-{-3ab6 + b*.
It contains f therefore^ the cubes of the two parts of the root, and
beside that* Saab + Sabb,2i quantity equal to (3 ai) x(a + b);
that is, the triple product of the two parts, a and h, inuUiplted by
their sum.
293. So that whenever a root is composed of two terms, it is
easy to find its cube by this rule. For example, the number
5 = 3+2; its cube is therefore 27 + 8 -{- 18 x 5 = 125.
Let 7 + 3 = 10 be the root ; the cube will be
343 + 27 + 63 X 10 = 1000.
To find the cube of 36, let us suppose the root a6 = 30+6,
and we hare for the power required,
--^ 27000 + 216 + 540 X 36 = 46656.
294. But if, on the other hand, the cube bo given, namely,
a* + Qaab + Sabb + b^, and it be required to find its root, we
must premise the following remarks :
Fii^t^when the cube is arranged according to the powers of
one letter, we easily know by the first term a^, the first term a
of the root, since the cube of it is a' ; if, therefore, we subtract
that cube from the cube proposed, we obtain the remainder,
Saab + 3abb+b^, which must furnish the second term of the
root
295. But as we already know thai the second term is + b,
we have principally to discover how it may be derived from the
aboTe remainder. Now that remainder may be expressed by
two factors, as (3 a a + 3 a 6 + b () x (1) ; if> therefore, we divide
■» N
Cfhap* 9. Of Comfmifd ((uantUits. 95
by Saa + Sah+bb, w« obtain the sec6ndpart of the root +6»
which 19 required.
296. But as this second term is supposed to be unknown, the
divisor also is unknown ; nevertheless we have the firat term of
that divisor, which is sufficient | for it is S a a, that is, thrice the
square of the first term already found ; and by means of this, it
is not difficult to find also the other part, fr, and then to complete
the divisor before we perform the division. For this purpose,
it will be necessary to join to S a a tlirice the product of the two
terms, or 3 a ft, and b b, or the square of the second term of the root.
£9r. Let us apply what we have s^d to two examples of other
given cubes*
I. a^ +12aa + 4Sa + M(a + 4
Saa-f.12a-f.l6) lSaa + 48a4.64
12aa+48a-f.64
0.
II. a«— .6a» + 15a*—S0a*+ I5a*-^6a +1
«• (a« — 2tf + l
Sa«-»6a> +4aa) —6a' + 15a« — 30a>
— 6a* + 12a^— 8aS
Sa^-«12a"+l2aa+3a>— .6a + l)Sa«— .]2a' + 15aa-.6a4-l
Sfl*— 12a» + 15afl— 6a + l
0.
298. The analysis which we have given is the fomidation of
the common rule for the extraction of the cube root itt numbers.
An example of the operatioa in the number 2 19r:
2197 (10 + S = 13
1000
300 1197
90
9
399 1197
0.
96 MgdnxL.
Let us also extract the cube root of 3496578$ :
Sect. 2.
34965783 (300 +20 + 7
27000000
270000
18000
400
7965783
288400
5768000
307200
6720
49
2197783
313969
2197783
0.
CHAPTER X.
Of the higher Powers of Compound Quantities.
299. Aftbb squares and cubes come higher powers^ or
powers of greater number of degrees. They are represented
by exponents in the manner which we before explained : we
have only to remember, when the root is compound, to inclose
it in a paranthesls. Thus (a + b)' means thata + b is raised
to the fifth degree, and (a — b)^ represents the sixth power of
a-^b. We shall in this chapter explain the nature of these
powers.
300. Let a + & be the root, or the first power, and the higher
powers win be found by multiplication in the following manner :
Cbap. 10. Of Omfcmi (^mmISI&u. 97
(a + 6)» a. fl + 6
a + 6
\
(a+6)».
+ ab + bb
a +fr
+ 006 + ^abb + b^
(«+»)*=
= o^ + $ao6 + do66 + fr3
a« + 30^6 4- 3aai6 + ah^
+ o»fr + 3flfl66 + 3o*» + 6*
(«+ ^)1'
o +*
fl5 4. 4fl46 4. 6a3W + 4aa6s + 06*
+ a^5 + 4aHb + 6006' + 4a6« + 6<
(fl+6)*
a=o» + 5o*6 + lOo^^d + 10flo6» + 5a6* + 6'
+fr ^
fl» + 5fl«6 + 10o*W + 100363 + 5aah^ + ab^
+ a*6 + 5a*W + lOo^fes + 10006* + Sah^ + 6«
(a+6)« «=s o« + 6o'6 + 150*66 + 200^6^ + 15oo6* + 606* + 6«
301. Tite powers of the root a— « ft are found in the same
manner^ and we shall immediately perceiTo that they do not
differ from the precedinq^, cxccptini; that the 2dy 4tb9 ^th^ &c.
terms are affected by the sign minui ;
Ed. JUg. IS
/ ..
#
98 MgArH. SeotS.
(o— J)»=o— »
a — b
1
1
aa
(o-
o
— b
o»
— iiaub+abb
— aab + Zabb-
-ft*
(«-
a
— Saub + aabb^
— b
i> ^ ■ ii
(a — b)*^=a* — 4(i>& + 6aaM — 4aft*+6«
a —b
— a^b + 4a^bb — 6aab^ +4ab^ — 6*
(o— J)»=a* -^ 5a** + 10a*66— lOoo** + 5ab* — b*
a—b
— a*6+ 5a-*W — I0a»*»+10aa6* — 5aft* +6*
(a— &)•= a« — 6a»6 + \5aHb — 2()a»6« -f- ISooJ* — 6a6» + 6».
Here we see that all the odd powers of <|> have the sign •— ^
while the even powers retain tbb si^ +. TM reason, of this is
evident ; for since — 6 is a term of the root, the powers of that
letter will ascend in the fpllowing series, — 6, + M, — b*f + b^,
— 6*9 -f- ^.. &c. which clearly shews that the even powers must
be affected by the sign +p and the odd ones by the contrary
sign — .
Bbapb IB. Of Comptmni (fuaiHUHes. 9&
S02. An important question occurs in this place ; namdyf how
we may find* without being obliged always to perform tbe same
calculation, all the powers either of a -f (» or a i— &•
We must remark, in tbe first place, that if we can asmgn all
tbe powers of a -f b, those of a — ft are also found, since we
have only to change the signs of the even terms, that is to say,
of the second, tbe fourth, the sixth, &c. The business then is
to establish a rule, by which any power ^ a -f b, however high,
may be determined vnihaiU the neeesriiy of eakuloHng all the pre»
ceding ones.
SOS. Now, if from the powers which we have already deter-
mined we take awi^ the numbers that precede each term* which
are called the coefficients, we observe in all the terms a singular
order ; Jirstf we see the first term 9^ of the root raised to the power
w%ieh is required ; in the following terms the powers of a diminish
continaUy by unity^ and the powers of b increase in the same
proportion ; so that the sum of the escponents of a and ofhis
always the same, and always equal to the exponent cf the power
required ; and, lastly, we find the term b by itself raised to the
same power. If, therefore, the tenth power of a + b were
requireii, we are certain that the terms, without their coefficients
would succeed each other in the following order ;^^^, a^b, a^b*,
an^, o«6*, a'b', a^b\ fl»^^ a«6S fl6% 6»*.
S04. It remains, therefore, to shew how we are to determine
the coefficients which belong to those terms, or the numbers by
which they are to be multiplied. Now, with respect to the first
term, its coefficient is always unity ; and rvith regard to the
second, its coefficient is constantly the exponent of the powers but
with regard to the other terms, it is not so easy to observe any
order in their coefficients. However, if we continue those coeffi-
cients, we shall not fail to discover a law, by which we may
advance as far as we please. This the following table will shew.
100 Mgdnn. " S«Gt.t«
Powers. Coefficients.
I. * 1, 1
II. 1, 2, 1
III. 1, 3, 3, 1
IV. 1, 4, 6, 4, I
V. 1,5,10,10,5,1
VI. 1, 6, 15, 20, 15, 6, 1
VII. 1, 7, 21, S5, 35, 21, 7, 1
VIII. 1, 8, 28, 56, 70, 56, 28, 8, 1
IX. 1, 9, 36, 84, 126, 126, 84, 36, 9, 1
X. 1,10, 45, 120, 210, 252, 210, 120, 45, 10, 1, &C.
We see then, that (he tenth power of a -f- 5 *vi!l be a^ ^ -f
10a»i +45a«66 +J20a^63 ^ 2Ho«6* + 25Ja'6» +210o*6« +
120a«6^ +45aaft» +10aJ» +6*«.
305. With regard to the coefficients, it must be observed, that for
each power their sum must be equal to the number 2 raised to the
samt power. Let a= 1 and b^l, each term, without the
coefficients, will be = 1 ; conseqdently, the value of the power
will be simply the sum of the coefficients ; this sum, in the pre-
ceding example, is 1024, aiid accordingly
(1 +l)^« = 2»« = 1024.
It is the same with respect to other powers ; we have for the
L 1+1=2 =2S
II. 1+2 + 1 = 4 = 2%
III. 1 + 3 + 3 + I = 8 = 2»,
IV. 1+4 + 6 + 4 + 1 = 16 = 2*,
V. 1+5+10+10 + 5 + 1=32 = 2*
VI. l+6+15 + 20 + 15+6 + l=64 = 2«
Vir 1 + 7 + 21 + 35 + 35 + 21 + 7 + 1 = 128 = 2% &C.
306. Another necessary remark, with regard to the coeffi-
cients, IS, that they increase from the beginning to the middle,
and then dcci*ease in the same order. In the even powers, tho
greatest coefficient is exactly in the middle ; but in the odd
powersn two coefficients, equal and greater than the others, arc
found in the middle, belonging to the mean terms.
The order of the coefficients deserves particular/ iattention ;
for it is in this order that we discover the means of determining
them for any power whatever, without calcidating all the pre-
\C
10. 0/ Cowfw$id (fmnliHes. 101
ceding powers. We shall explain this method^ reserving the
demonstration however for the next chapter.
307. In order to find the coeffidents of any power propoud, the
seventhi far example, let im write the foUawingfractionSf one after
the otlier ;
1 • i 4 S. 1. 1
T> T» Tf* T» T» T> y
In this arrangement we perceive that the wwmerators begin by tlie
eocpofwnt of the power regictredy and thai they diminish successi'oely
hf unity ; whUe the denominatprs follow in the natural order of the
numbers, 1,2,5, 4, S^c. M'ow, the first coefficient being idways I,
the first fraction ^es the second coeffident. The product of the
two first fractions, multiplied together, represents the tliird coefficient.
The product of the three first fractions represents tlie fourth coeffi*
eient, and so on.
So that the first coefficient = 1 ; the second = | = r j the
third = I X 1 = 21 ; the fouHh = | x | X | = i'S | the fiflh
= T^I>c4xi = 35 ; the sixth =|xlx4X$X| = 21;
the seventh =:2lxf=7'> the eighth = 7 x y = 1*
308. So that we have, for the second power, the two fractions
4f \ ; whence it follows* that the first coefficients L^ the second
= 1=2; and the third =2X 1 = 1.
The third power furnishes the fractions 4» l^ i ^ wherefore
tho first coefficient = 1 ; the second = | = 3 ; the third
= 3 X I = 3 ; the fourth =4x|x4 = l.
We have for the fourth power* the fractions ^, |f |f i ; con-
sequently the first coefficients 1 j the second 4 = 4 ; the third
« X I = 6 ; the fouHh ^ X | X | = 4 ; and the fifth 4 X | X $
Xi=l.
309. l^his rule evidently renders it unneccssaiy for us to find
the preceding coefficients* and enahles us to discover imme-
diately the coefficients which belong to any nower. Thus* for
the tenth power* we write the fractions y * ^ |, J, •, |, *, |i «,
tV» by means of which we find
the first coefficient = 1*
the second = ^-^ =10*
the third = 10 x { = 45*
the fourth = 45 X f = 120,
the fifth =120x^=210,
Ilie AfMl s= 210 X t s tBt,
the serenth ss 25£ X f = 210^
the eighth rs 210 x 4 = 120,
the ninth s=120x{s:45,
the tenth s 45 X | = lOy
the eleventh = 10 X^V^^ ^*
SIO. We may also write these fractions as they are, withoai
compating their value ; and in this way it is easy to espress
any power of a -f 6, however hi|^h. Thus, the hondredth power
of a + 6 will be (a + 6)»»« =ai«o + *«• x a««ft+ !5?_x^»
, IOO,x 99,X 98. 3 100x99x98x97, ,
^ ^ 1X2X3 ^1X2X3X4 "*'•»■#
&c.. whence the law of the sacceeding terms may be easi^
deduced.
CHAPTER XI.
Of the TransporiHoH rf the LetterSf m which the detnaiutipaium of
the preceding rule is f minded*
311. If we trace back the origin of the coefficients which we
have been considering, we shall find* that each term is presented,
as many times as it is possible to transpose the letters, of which
that term consists ; or, to express tlie same thing dilTerently,
the coefficient of each term is equal to the number of transposi-
tions that the letters admit, of which that term is composed. Tn
the second power, for example, the term a b is taken twice, that
is to say, its coefficient is 2 ; and in fact we may change the
order of the letters which compose that term twice, since we
may write a b and ha; the term a a, on the contrary, is found
only once, because the order of the letters can undergo no
change, or transposition. In the third power ot a + bf the
term a a( may be written in three different ways, a a 6, aba,
baa; thus the coefficient is 3« Likewise, in the fourth power, the
term a' 6 or a a a (, admits oFfour different arrangements, aaabf
aaha^ abaOf baa a; therefore its coefficient is 4. The term
a abb admits of six transpositions, a abb, abba^ babOf abab^
5 5a^^|( a b, and its coefficient is 6. It is the same in all cases*
Cha|i. 11. OJ Ctmfmtud qmoMHes. so 103
512. In fact» if we consider that the fourth powei^ for ezample»
of any root consisting of more than two teras^ as (a^- ftH- c +<Q*f
is found by multiplying the four factors, L a + b-^c + d;
11. a + b + c+d; III. a + b + c + d; IV. o + * + c + d ; we
me may easily see, that each letter of the first factor must be
multiplied by each letter of the second, then by each letter of
the third, and, lastly, by each letter of the fourth.
Each term must therefore not only be composed of four letters,
but also present itself, or enter into the sum, as many times as
those letters can be differently arranged with respect to each
other, whence arises its coefficient.
313. It is therefore of great importance to know, in how
many different ways a given number of letters may be arranged.
And, in this inquiry, we roust particularly consider, whether
the letters in question are the same, or diflbrent. When they
are tbe same, there can be no transposition of them, and for this
reason the simple powers, as a', a', a*, &c., have all unity for
the coefficient.
314. Let us first suppose all the letters different ; and begin-
ing with the simplest case of two letters, or a t, we immedi-
ately discover that two transpositions may take place, namely,
a b and b a.
If we have three letters a 5 c, to consider, we observe that
each of the three may take the first place, while the two others
will admit of two transpositions. For if a is the first letter, we
have two arrangements, abCfOeb; if ft is in the first place, we
have the arrangements bac^bea ; lastly, if c occupies the first
place, we have also two arrangements, namely, isab^cba. And
consequently the whole number of arrangements is 3 x 2 = 6.
If there are four letters, abed, each may occupy the first place;
and in each case the three others may form six different ar-
rangements, as we have just seen. The whole number of
transpositions is therefore 4x6 = 24=:4x3.x2x 1.09
If there are five letters, abcdCf each of the five must be the
first, and the four others will admit of twenty-four transpo-
sitions ; so that the whole number of transpositions will be
5X24 = 120 = 5X4 X 3X2X l.>* i
315. Consequently, however great tbe number of letters may
be, it is evident, provided they are all different, that we may
1{M ^K^Ara. Sect 2»
easily detemine the ttvoiber of transpositioiiflf and that we may
wake use of the roUowing table :
Nuipber of Letters. Number of Transpositions.
v_: ^ f y ^^.— 1 /
I. 1 = !•
II. SXl=2.
HI. 3x2x1 = 6.
IV. 4 X S X 2 X 1 = 24.
V. 5 X 4 X 8 X 2 X 1 = 120.
VI. 6X5X4X3X2X1 = r20.
VII. rx6x5x4x3x2xl = 5040.
VIII. 8xrx6x5x4x3x2Xl= 40320.
IX. - 9x8xrx 6X5X4X3X2X1 = 362880.
X. 10X9X8X7X6X5X4X3X2X1 = 3628800.
316. Bat, as we have intimatedf the numbers in this table
can be made use of only when all the letters are different ; for
if two or more of them are altkoy the number of tranqioeitions
becomes much less ; and if all the letters are the 8ame» we have
only one arrangement We shall now see how the numbers in
the table are to be diminished, according to the number of letters
that are alike.
317. When two letters are given, and those letters are the
same, flie two arrangements are reduced to one, and conse-
quently the number, which we have found above, is reduced to
the half ; that is to say, it must be divided by 2. If we have
three letters alike, the six transpositional are reduced to one ;
whence it follows that the numbers in the table must be divided
by 6 = 3 X 2 X 1* And for the same reason, if four letters are
alike, we must divide the numbers found by 24 or 4 x 3 x 2 x It
&c. ' ,
It is easy therefore to determine how many transpositions the
letters aaabhCf for example, may undergo. They are in number
6, and consequently, if they were all different, they would
admit of 6x5x4x3x2x1 transpositions. But since a is
found thrice in those letters, we must divide that number of
transpositions by 3 x 2 x 1 ; and since 6 occurs twice, we roust
again divide it by 2 x 1 ; the number of transpositions required
.11 ^L r u 6X5X4X3X2X1 ^ ^^ „ ^^
wUl therefore be = — - — -r — ; — - — -— = 5 x 4 x 3 = 60.
3X2X1X2X1
Chap. II. Of Compaufid qiumiiHes. 105
318. It will now be easy for us to determtne the eoeflfcients
of all tlie terms of any power. We shall give an example of the
seventh power (a + hy.
The first term is a^, which occurs only once ; and as all the
other terms have each seven letters, it follows that the number of
transpositions for each term would be7 x 6x 5 x^ X^ X^X I,
if all the letters were different But since in the second
term, a^hf we find six letters alike, we must divide the above
product by6x5x4x3x2Xly whence it follows that the
OB. .' 7x6x5x4x3x2x1 .
coefficient is = — ^ :: :; r— = t»
6x5X4x3X^X1 *
In the third term a' b b, we find the same letter a five times^
and the same letter b twice ; wo must therefore divide tHat
number first by5x4x3x2xiy and then also by 2 x 1 ;
whence results the coefficient ! ^ ^ ^ ^ ^^^^ ^^^ ! = P^.
5 X4 XiXiiX 1 XiJX I 2X1
The fourth term a^ b^ contains the letter a four times, and the
letter 6 thrice ; consequently^ the whole number of the transpo-
sitions of the seven letters mMst be divideil, in the fii'st place,
by 4 X 3 X 3 X If and secondly, by 3 x 2 x if and the coeffi-
. ., 7x6x5x4x3x2x1 7x6x5
cient becomes =
"•
4X3X2X1X3X2X4 I X 2x3
7x6x5x4
In the same manner, we find — - — 7 for the coefficient
1x2x3x4
of the fifth term, and so of the rest ; by which the rule before
given is demonstrated,
319. Th^e considerations carry us further, and shew us also^
how to find all the powers of roots composed of more than two
terms. We shall apply them to the third power of a-f-6 -f-c j
the terms of which must be formed by all the possible combina-
tions of three letters, each term having for its coefficient the
number of its transpositions, as a^ove.
Without performing the multiplication, the third power of
(a + b + c) will bea'-f-3aab-f.3aac-{-3abb + 6a6c-f3acc
-f t ^ -f- 3 6 6 c + 3 6 c c + c 3 .
Suppose a == 1, 6 =: 1, »= 1* the cube of 1 4- 1 4. 1, or of 9*
will be 1+3+3 + 3 + 6 + 3 + 1 + 3 + 3 + 1= 27.
This result is accurate, and confirms the rule.
EuL Mg. 14
ft
106 JKlg/ikra. Skci.%
If we had flUHiosed a = l, & = !» ande=s — ly wa shodd
iMive found for the cabe of 1 + 1 — 1» that Is, of 1,
1 4-5 — d + 3 — 6 + 3 + 1 — 34.3^1 si.
CHAPTER XII.
Of the expremon of Irrational Powers bylnfinUe Series.
320. As we have shewn the method of finding any power of
the root a +b, however great the exponent^ we are able to
express* generally^ the power of a + fry whose exponent Is unde-
termined. It is evident that if we represent that exponent by
n, we shall have by the role already given (art 307 and the fol-
lowing) : ^
(a + 6)» = a" +- «»-» 6+ - X — r— o"-*6» +-r X — 3— X
n — 2 ^ -._ n n — 1 n— 2 n — 3 „ ^»ac.
__ tt'-**^ + _ ^ __ X —^ X -^ a" -^ 6* &c.
321. If the same power of the k*oot a-— b were required, we
should only change the signs of tli^ second, fourth, sixth, &c.
H If II "™* 1
terms, and should havc(a — fc)*sf:a*— -rO*-*i +— x — 3—
t.-«x« * n— Jf n — 8 ^ ,., . II n— 1 11 — 2_
— - — a"~^6S&c.
4
322. These formulas are remarkably useful ; for they serve
also to express aU kinds of radicals. We have shewn that all
irrational quantities may assume the form of powers, whose
exponents are fractional, and that \/a =0^; \/a zso^, and
4_ 1
v^a = d^f kc. We have therefore, also,
V(a + 6) = (a + ft)^; \/(«+*) = C« + 6>^
4 1
and v' (a + 6) = (a +*)^> &c.
Wherefore, if we wish to find the* square root of a + &, we
have only to substitute for the exponent n the fraction 4» in the
general formula, [art. 320,] and we shall have first, for the
coefficients,
Chap. 12. (Jf Compound Quantilies. 107
n 1 n — 1 1 n — 2 S n — 3 5 n — 4
• -^ —^ •
1"" 2' 2 "" 4' 3 "" 6' 4 ^ b' 5
— --;^^ =— — . Then, a" =a^=V^anda""* = -7=;
10 O 12 \/a
a""'= — =; a*""*=: =, &c., or we might express those
ax/a ' - aa\/a or
_ a*
powers of a in the following manner; a'sv'a ; a*"^ = — =
T"^^ "^-7?^''* -;^-^r»« -r"-"i^'*^'
323. This being laid down, the square root of a + A, may be
expressed in the following manner :
V (« + «) =
— itVflT 1 iktV<r . 1 1 3 . , vi" 1 1
m
324. If a, therefore, be a square number, we may assign the
value of y^^ and, consequently, the squara root of a + i may be
expressed by an infinite series, without any radical sign*
Let, for example, a=2cc, we shall have y^^ = c ; then
I b Ihb I h^ 5 M .
V(e see, therefore« that there is no number, whose square
root we may not extract in the same way ; since every number
may be resolved into two parts, one of which is a square repre-
sented by cc If we require, for example, the square root of 6,
we nk^ke 6 s= 4 + 2, consequently c c = 4, c = 2, 6=2, w hence
results V6: ^ 2 + 1 — ^V + T7 — t/t4> *^*
If we tal^e only the two leading terms of this series, we shall
have 2| = (, the square of which, y, is ^ greater than 6 ; but
if we consider three terms, we have Sl^j = 479 the square of
which« VtV, is still ^y^ too smaiU
325. Since, in thi& example, | approaches very nearly to the
true value of ^6^ ^^ sball take for 6 the equivalent quantity
V — h 'J'hus c c = y I c = f 5 6 = — » ; and calculating only
the two leading terms, we find V6 = | + i X %- = | — | X -^
lOS MgAra» Sect. £•
ss I *-« fV = T7 : tbe square of this fraction^ being y/^^ , exceeds
the squall of x/S'only by j^.
Now, making 6 = %V — ' isU* ^ that c = J • and 6 = — ^^ J^ j
and stili taking only the two leading terms, we have
— * 1
^/r- — A9 f ly ?>g — 49 lv^7inr^49 t 480 1
iv it
the square of which is \^^^/^. Mow 69 when reduced to the
same denominator, is = «^YtVtV 5 t**® error thei-efore is only
326. In the same manner, we may express the cube root cf
3 1
a + 6 by an infinite series. For since v (a + () = (a + 6)^#
we shall have in tbe general formula n = ^, and for the coeffi-
cients, -j---;"-^ — -; -^«— -; -j- -^ ;
"7 = — Tk^ &c., and with regard to tbe powers of a, we shall
3__ 3^ 3 _
havca» = v^a»"l = ^;af-*=:5^|tf^3fc^&c.; then
a - aa /a
i'-^^ = v.-+ix.^-i9^.^+i^*' ^
a*
243 a*
327, If a therefore be a cube, or a = c', we have \^a = c, and
the radical signs will vanish ; for we shall have
328. We have, therefore, arrived at a formula, which will
enable us to find by approximation, as it is called, the cube root
of any number ; since every number may be resolved into two
parts, as c^ +0, the first of which is a cube.
If we wish, for example, to determine the cube root of 2, we
represent 2 by 1 + 1, so that c = 1 and ( = 1, consequently
3
V2 = 1 + 7 -**- 7 + TT' ^^*^ t''^ t^^ leading terms of this
Chap. IS. Qf Compound quantities* IQS
series make H = ^ the cube of which, |4« is too great by 4|.
Let us then make 2 = fy — |t» we have c = 4 and 6 = — j^,
3__ -10.
and consequently va = y + y X -^. These two terms give
T — t\ = t4' **»e cube of which is 14|f U- Now, 2 = mt^f
so that the error is ^rfl^T- In this way we might still approx-
imate, and the faster in proportion as we take a greater number
of terms.
CHAPTER XIII.
Of Vie resolutian of Jflsgative Powers.
329. We have already shewn, that we may express — by ar^ ;
we may therefore also express --r-T by (a + ()^^ ; so that the
fraction — -: may be considered as a power of a + (, namely,
that power whose exponent is — 1 ; and from this it follows,
that the series already found as the value of (a + h)* extends
also to this case.
SdO. Since, therefore, —-r is the same as (a-f- b)*S let us
suppose, in the general formula, n= — I; and we shall ''rst
have for the coefficients — = — 15 "" ■ = — 1 j ~ = — 1 ;
fi— 3 1
— - — s= — 1, &c. Then, for the powers of a ; o» = a"* = — ;
4 a
1 * 1 1
ft«~» = a-*= -; a"-*= T ; a"-* = ,, &c. So that (a + b)-^
1 ^ ^ ,f^b b\ b* b^ .
= -7-1= i+-i — -1+ "1 — ■«» *c-» *"** this IS the
tt+b a fl' o* a* * a' o*
same series that we found before by division.
S31. Further, being the same with (a + 6)-*, let us
reduce this qnantity also to an infinite series. For this purpose,
we must suppose n = --*- 2, and we shall first have for the coeffi-
ltd JUgAra. Sect s«
5 II
, &c. Then, for the powers of o: a"= -=-; a""* = -? 5
4 a' ' a' '
«*-* = -i ; «*^ = -yt *c. We therefore obtain (a + A)-* c?
|x| = 4^ T^l^4x|^59&c. Conseqiieiitlyy we ha?e
S5sis. Let us proceed and suppose n = — 3» and we shall have
a series expressing the value of ^ tu\^ 9 or of (a + 6)-^. The
coefficients wril be ~ = jj —_ = __. __-=_.-;
^. *" ■■ = 7, &c. and the powers of a become, a" = -i ; a»^* =
-s J a*^ = -if &c., which gives , — r-rrr = -^ — -r -r H —
4 5» S 4 5 6» 3 4 5 ^**i,
^"i i^^T^l^l^+l^ 2^"3^4 J7**^-
1 ft ft* 6^ 64 55 5« ^T
a**
Let us now make it s — • 4 ; we shall have for the coe2cients
n 4 n— i_ 5 n— 2 _^ 6 n— 3_ 7 <
T-""I' "2 ~1' "T"""""^* "^""""~7 ^^'^
and for the powers^ a« =s -^ ; a""* = — ; a"^ == — ; a"-* = -— 5
(I* A* a fl*
o-*=i,&c., whence we obtain J ^^-p^ = ^ —* X jy +
45 6« 456 *» 4567" »• .
T^ 2^ r«--i><i^ »*««-? + T^i^ "3^4^ fl-yJ*"-
Chap, 13. Of C(mip(mndquantUies. Ill
I b hf b* b* h' .
333. The different cases that have been considered enable us
to conclude^ with certainty, that we shall have, generally, for any
negative power of a + b;
I __ 1 m b m m+l b* m in+ 1
«+2 . JI!L it
And by means of this fomnulaf we may transform all such
fractions into infinite series, substituting fractions also, or frac-
tional exponents, for m, in order to express irrational quan-
tities.
334. The following considerations will illustrate this subject
further.
We have seen that,
I I b b^ b^ b^ b'
= ; + -t i + "S k *^*^*
n. a* * ill #t4 ■ ill #■«
fl + ^ « «• a' o* a' a^
If, therefore, we multiply this series by a + 6, the product
ought to be = 1 ; and this is found to be true, as we shall see by
performing the multiplication :
1 b b^ b^ b^ M
a + fr
b fca 5» 6* ft* . £.
fl o» a' fl* a*
1.
335. We have also found, that
1 1 26 3ftft 46* ^_6ft^ fi^
If, therefore, we multiply this series by (a + ()*, the product
ought also to be = 1. Now (a + b)* z=zaa+Qab + bb. See
the operation :
lis MgAra. * Sectfi*
1 fift ^hb 46* , Sd^ 66*
2ft S66 463 554 6(,«
a ^ aa a^ ^ a^ o* ^^
26 466 66* 86* 106*
• a a a a* a^ a'
hb^ 26» S6* 46»
' aa a^ a* a'
1 = the productf which the nature of the thing required.
356. If we multiply the series which we found for the value
of ; , ,,- , by a + 6 only, the product ought to answer to the
fraction — ^— rt or be equal to the series already founds namely,
fl + 6
J + ~ 4 + "^' ^^* ^^^ *''*® ^^^ actual multiplication
wiU confirm.
1 26 366 4 6» 5b* ^
a a a* a* a' a^
a +b
1 26 366 46' 5 6* „
6 26 6 3 6» 4 6* „
4. J , &c.
^ aa a^ ^ 0* a* *
1 6 66 6» fc*
a aa "^ c3 a* ^ a^
'x
^«
SECTION m.
OV BATIOS iJID FBQPOETIOirS.
CHAPTER. I.
6f Arithmetical Ratio, or of ike difference between two Jfumhers.
Two quantities are either equal to one anotlfer» or they are
not. In the latter case, v^here one is greater than the othern
ive may consider their inequality in two diffei*ent points of view :
we day ask, hmio much one of the quantities is greater than the
other 7 Or, we may ask» how many times the one is greater
than the other ? The results, which constitute the answers to
these two questions, are both called relations or ratios. We
usually call the former arithnutical ratiOf and the latter geomet*
rical ratio, without however these denominatioas having any
connexion with the thing itself: they have been adopted arbi-
trarily.
338. It is evident, that the quantities of which we speak must
be of one and the same kind ; otherwise, we could not* determine
any thing with regard to their equality or inequality. It would
be absurd, for example, to ask if two pounds and three ells are
equal quantities. So that in what follows, quantities of the
same kind only are to be considered ; and as they may always
be expressed by numbers, it is of numbers only, as was men-
tioned at the beginning, that we ahall treat.
339. When of two given numbers, therefore, it is required to
find, how much one is greater than the other, the answer to this
question determines the arithmetical ratio of the twu i:umbei's»
Jfovff since this answer consists in giving the diffei*ence of the
EuL Jttg. 15
t.
114 Mgara. Sects.
two numberSf it followsf that an arithmetical ratie is nothing
but the difference between two numbers : and as this appears to
be a better expi^ssion, we shall reserve the words ratio and
relationf to express geometrical ratios.
340. The difierenre between two numbers is found* we know,
by subtracting the less from the greater ; nothing therefore can
be easier than resolving the question) how much one is greater
than the other. So that when the numbers are equals the dif-
iP^rence being nothings if it be inquired how much one of the
numbers is greater than the otliery we answer^ by nothing. For
example, 6 being = 2 x 3, the difference between 6 and 2 x 3 is 0«
341. But when the two numbers are not equal, as 5 and S,
and it is inquired how much 5 is greater than 3, the answer is,
2; and it is obtained by subtracting 3 from 5. Likewise 15
is greater than 5 by 10 ; and £0 exceeds 8 by 12. ^i " "
342. We have three things, therefore, to consider on thisr
subject ; 1st, the greater of the two numbers ; 2d, the less ; and
3d, the difference. And these three quantities are connected
together in such a manner, that two of the three being given,
we may always determine the third.
Let the greater number = n, the less = 6, and the difference
= (1; the difference d will be found by subtracting bhom a, so
that d = a — b; whence we see how to find d, when a and b are
given.
343. But if the difference and the less of the two numbers, or
(, are given, we can determine the greater number by adding
together the difference and the less number, which gives a =
b + d. For,, if we take from b + d the less number ft, there
remains d, which is the known difference* Let the less number
= 12, and the difference = 8 ; then the greater number will be
= 20.
344. Lastly, if beside the difference d, the greater number a
is given, tlie other number 6 is found by subtracting tlie differ-
ence from the greater number, which givesft=a — d. For if
I take the number a — d from the greater number a, there
remains d, which is the given difference.
345. The connexion, therefore, among the numbers a, b, df is
^of such a nature, as to give the three following results : l*^ d = a
Chiip*S« Of Batios ani Proportion, 115
— 6; 2^ asib + d; S^ frr=a— -cl; and if one of these three
comparisons be justj the others must necessarily be so also.
Wherefore, generally, if « = a: + y, it necessarily fallows, that
y =«— or, and ar=:s:) — y,
S46. With regard to these arithmetical ratios we must remark^
that if we add to the two wiimbers a and b, a number c assumed
at j^surCf or stibtrad it from them, the difference remains Hit
same. That is to say, if d is the difference between a and 6,
that number d will also be the difference between a +c and
h + c, and between a — - c and fr— - c. For example, the differ-
ence between the numbers 20 and 1<2 being 8, that difference
will remain the same, whatever number we add to the numbers
20 and 129 and whatever numbers we subtract from them.
347. The proof is evident ; for if a — ft = d we have also
(a + c) — (6 + e) = d ; and also (a — c) — (b-^ c):=zd.
548. If we double the trvo numbers a and b, the difference roiU
also become double. Thus, when, a — b = d, we shall have,
2a — 26= 2d; and, generally f na-^nb=:nd, whatever value
we give to n.
CHAPTER IL
Of Arithmetical Proportion*
349. When two arithmetical ratios, or relations, are equal,
this equality is called an arithmetical proportion.
Thus, wlien a — 6 = d and p — g = d, so that the difference
is the same between the numbers p and q, as between the num-
bers a and 69 we say that these four numbers form an arithmeti-
cal proportion ; which we write thus, a — 6 =p — q, expressing
clearly by this, that the difference between a and b is equal to
the difference between p and q.
350. An arithmetical proportion consists therefore of four
terms* which must be such, that if we subtract tiie second from
the first, the remainder is the same as when we subtract the
fourth from the third. Thus, the four numbers 12, 7, 9, 4, form
an arithmetical proportion, because 12 — 7=:9.^4«*
• To shcr that these terms make such a proportion^ some write them
thus ; 12 • , 7 : : 9 . . 4.
11« Sgthrm. Seet.l4
S&1. When, we him an ariihmdiM )mpdftioii, as a^— basy
^-^ q* tc^e may make the sectmd and third change placest wriinig
H — psb — q; and this equality wiU be ne lesstme; fory aiiic^
a — b=zp — Qf add (to both sides^ and we have a = fr+p ^^9f
then siibtractp from both sideSf and we havd a-*—p=:fc—- >}•
In »he same manner) as 12 — 7 s 9 •** 4» so also
12 — 9 = r — 4.
352. We may 9 in every arithmeiickU proportimif jmt the secmid
lerm also in the p(nce of Ute Jirst^ y we make the same iramspati^
Hon of the third and fourth. That is to say, if a — 6 = p <— » f^
we have also b — a = 9-^ p. FbrA — n Is the negative of
a — • 69 and q*^p is also the negative of p — g. ThuSf sinee
12 — 7 = 9 — 4, ^e have also 7 — 12 =±:4 —• 9.
353. But the great property 'of eroery arithmetieal proporHoni8
this ; that the sum of the second and third term is always equal to
t/ie sum (f the Jirst and fourth. This piiipert j, which we must
pailicularlv consider, is expressed also by saying that, the sum
of the means is equal to the sum of the extremes. Thus, since
12 — 7 = 9 — 4, we have 7 + 9= 12 + 4; and the sum we find
is i 6 in both.
354. In order to demonstrate this principal property, let
a — 6=p — 9; if we add to both 6 + 9* we have a + q::^b+p;
that is, the sum of the first and fourth terms is eqaal to the sum
of the second and third. Jnd corrcersely^ if four vumbers^ a, b, p>
q, are stichf tJuit the sum of the second and tAtrd is equal to Hie sum
of the Jirbt and foiirtkf that Is," if ft + p = a + g, we conclude^
without a i»ossibility of mistake, that these numkers are in ariih"
metical proportion^ and that a — 6 = p — g. For, since
a + g=6 + p,
if we subtract from both sides ft + g, we obtain a — ft=p — g.
* Thus, the numbers 18, 13, 15, 10, being such, thaH; the sum
of the means (13 -f 15 = 28,) is equal to the sum of the ex-
tremes (18 +M0 = 28.) it is certain, that they also form an
arithmetical pr(»portion ; and. consequently, that
18—13 = 15—10.
355. It is easy, by means of this property, to resolve the fol-
lowing question. The tliree first tera s of an arithmetical pro-
portion being given to find the fourth 7 Let a, ft, p, be the three
\
ebMf.lt. •- Of RoHm md Proportiofu IIT
first termsy and Ua m express the fourth hj f, which it in
required to determiiief then a-f9=sft4*p; by subtracting •
from both sides, we obtain q=zb 4. j» — a.
. Thast ike fourA term is found by addwg iogelktr the second and
thirdf and suttraehng the first from that sum. Suppose^ for ex-
auiple» that 19» SSy 13, are the three first terms given^ the sum
of the second and third is =41 ; take from it the first, which is
19» there remains iStSor the fourth term soughtf and the arith-
metical proportion will be represented by 19 — 28 = 13 — Q2$
or, by £B — 19 = 22— 13, or, lastly, by 28 — ^ = 1 9— 13.
856. When in an arithmetical prcpartion, the seamd term is equrd
to the thirdf we have atdy thru numbers ; the property of which
is this, that the first, minus the second, is equal to the second,
minus the third $ or, that the^ difference between the first and
the second number is equal to the diflference between the second
and the third.' The three numbers, 19, 15, U, are of this kind,
since 19 — 15 s 15 — II.
357. Three sucA nwmbers are said to form a continued arith"
metioal pnspoHion, which is sometimes written thus, 19 : 15 : IK
Such proportions are also called arithmetical progressions, par^
tkularty if a greater number of ter^fdkno each other according
to the same law.
An arithmetical progression may be either increasing^ or
decreasing. The former distinction is applied when the terms
go on increasing, that is to say, when the second exceeds^the
first, and the third exceeds the second by the same quantity ;
as in the numbers 4, 7, 10. The decreasing progression is that,
in which the terms go on always diminishing by the same quan«
tity, such as the numbers 9, 5, 1.
258. Let us suppose the numbers a, 6, c, to be in arithoic^ticai
progression ; then a — 6 = A — • c, whence it follows, from the
equality between the sum of the extremes and that of the means,
that 2 b =: a + c,- and if we subtract a froorbothp we have
c=:2b-~a«
359. So that when the two first terms a, b, of an arithmetical
progression are given, the third is found by taking the first from
twice the second. Let 1 and 3 be the two first terms of an arith-
metical progression^ the third will be=:£>c3— 1 = 5. And
these three nuihbers 1, 3, 5 give the proportion 1 — 3 = 3 *^ 5.
118 JKgdnra. fieef. &
360. By following the same fndfaoff, we maj pursue the
arithmetical progression as far as we please ; wt have onbi to
find thefimrth by means cfthe second and third, in the same man*
ner as we determined the third by means of the first and second,
and so on. Let a be the first termf and b the second, the third
Will be=:2fr — a, the fourth =4( — 2a — fr=:dfr~£a, the
fifth 65 — 4a — 2b + a:=i4b — Sa, the sixth =:8ft — 6a— 3i
+ 2a=:5b — 4a» the seventh =10fr — 8a — 4A + 3a = 66
— 5 a, &c.
CHAPTER III.
Of ^rithnielical Progressions.
361. TTe have remarked already, that a series of numbers
composed of any number of terms, which always increase, or
deci*ease by tlie same quantity, is called an arithmetical progres*
snon* 4_
Thiis, the natural numbers, written in their order, (as 1, 2, S,
4f 5, 6, 7, 8, 9', 10, &c.) form an arithmetical progression^
because they constantly increase by unity ; and tlie series £5,
£S, 19, 16, 13, 10» 7f 4, 1, &c. is also such a progression, since
the numbers constantly decrease by 3.
362. The number, or quantity, by which the terms of an
arithmetical progression become greater or lesa^ is called the
difference. So that when the first term and the difierence are
given, we may continue the aritlimetical progression to any
length.
For example, let tlie first term =: % and the difierence = S^
and we shall have the following increasing progression ; 2, 5,
8f llf 14, 17, 20, 23, 26, 29, &c. in which each term is found,
by adding the difierence to the preceding term.
363. It is usual to write the natural numbers, 1, 2» 3, 4, 5, &c.
above the terms of such an arithmetical progression, in order
that we may immediately perceive the rank which any term
holds in the progression* These numbers written above the
■>
V /
S
Chap. S. ' Of Maiios and Proportion. 118
terms^ may be called iniSiAB ; and the above example is written
as fdllows :
Indices^ 123 4567 89 10
JirUh. Prog. 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, &c.
where we see that 29 is the tenth term.
364. Let a be tlie first term, and d the diflference, the arith-
metical pragression will go on in the following order :
1 ; 2 3 4 5 6 7
a, a'+d, a + ^d, a + 3d, a + 4rf, a + 5df a + 6d, &c.
whence it appears, that any term of the progression might be
easily found, without the necessity of finding all the preceding
ones, by means only of the first term a and the difference d.
For example, the tenth term will be = a -f- 9 d, the hundredth
term = a + 99 d, and generally, the term n will be
=:a + (n — l)d. ^
365. When we stop at any point of the progression, it is of
importance to attend to the first and the last term, since the
index of the last will represent <Ae number of terms. ijT, there"
fore, the first term = a, the difference = d, and the number of ierm$
= n, tve shall have the last term = a -f- (n — 1) d, which is con-
sequently found by multiplying tlie difference by the number of terms
minus one^ and adding the first term to that product. Suppose, for
example, in an arithmetical progression of a hundred terms,
the first term is = 4, and the difibrence = 3 ; then the last term
^ IV ill be = 99 X 3 +4 = 301.
366. Vben^we know the first term a and the last %9 with the
number of terms «n, we can find the difference d« For, since
the last term %=ia + (n — 1) d, if we subtract a from both sides,
we obtain oi — at=(n — 1) d. So that by subtracting the first
^rm from (Tie fast, we have the product of the difibrence multi-
plied by the number of terms minus 1. We have, therefore,
only to divide » — a by n — 1 to obtain the required value of
the difference d, which will be = ^ — . This result furnishes
II — 1
the foljowing rule : fifubtract the first term from the last, divide the
remainder by the number of terms minus 1, and the quotient vnll
be the difference : by means of which we may write the whole
progression. ,
ISO MgOm. Sect A.
367. Suf^KMCy for example, that we have an arithmetical
progression of nine terms, whose first is =: '2, and last =s S6»
and that it is required to find the difference. We must subtract
the first term, 2f from the last, 26, and divide the remainder,
which is 24, by 9 — 1, that is, by 8 ; the quotient d will be equal
to the difference required, and the whole progression will be
1234 56789^
8» 5, 8, 11, 14, 17, £0, 23, 26.
To give another example, let us suppose that the first term
= 1, the last = 2, the number of terms = 10, and that the arith*
metical progresiBion, answering to these suppositions, is requir-
2-— I I
ed ; we shall immediately have for the difference yr — - = -^,
and thence conclude that the prog ression is
1 2^8 4 5 6 7 8 9 10
1, 1^, 1}, 1|, If, If 1|, If if 2.
Mother example. Let the first term = 2^» the last =; 12|,
and the number of terms = 7^the difference will be
124^-2J_10|_6£_ 25
7— I "" 6 ""se"" 36*
and consequently the progression
1 2 3 4 5 6 7
Hy 4,V» ^ih Vt» 9|. 10|f 12|.
268. If now the first term a, the last term %, and the differ-
ence df are given, we may from them find the number of terms *
n. For since %^^a = (n — 1) (f, by dividing thi^ .two sides
hj dj we have — ^ — = n — 1. Now, n being greiter by 1
than n — 1, we have n = "T + 1 J consccjuently, the number
of terms is found by dividing the difference between thejtrst and Ike
lasttermtor z-*-a, Ay the difference of the progression, and adding
unity to the quotient, — ^ — .
For example, let the first term =4, the^Iast = 100, and the
100 — 4
difference = 12% the number of terms will be — rr 1. 1 = 9 ;
'and these nine terms will be,
123456789
4, 16, 23, 40, 52, 64, 76, 88, 100.
Chap. 3. Of Batios and Vrciportim. 121
If the first term s 2f the last = 6, and difference s 1^, the
number of terms will be -j + 1 c= 4 ; and these four terms
will be^
12 5 4
2, 3|, 4|, 6,
Againf let the first term =s S|, the last =s 7f f and the differ-
ence = 14, the number of terms will be = ^ /^ 4. 1 = 4 ;
It
which ara^
54, 4|, 6 J, r|.
369. It must be observed, however* that as the number of terms
is necessarily an integer, if we had not obtained such a number
for n, in the examples of the preceding article, the questions
Would have been absurd.
Whenever we do not obtain an integral number for the value
of —^ — 9 it will be impossible to resolve the question 5 and con-
sequently, in order that questions of this kind may be possible,
% — a must be divisible by i.
370, From what has been said, it may be concluded, that we
have always four quantities, or things, to consider in arithmetic^
al progression ;
L The first term a.
11. The last term it.
III. The difference d.
IV. The number of terms o.
And the relatiouH of tiiese quantities to each other are such, that
if we know three of them, we are able to determine the fourth 5
for.
I. If a, d, and n are known, we have z = a + (n — 1) d.
II. If z, d, and n are known, we have a = z — (n 1) d.
III. If a, z, and n are known, we have d =
z — a
z— a
IV. Tjf a, z, and d are known, we have n = — j — f. 1.
SiU. Mg. 16
L
1218^ JUgtbta, Sect ^
CHAPTER IV.
(tf {lit Summation of Jirithmetical Progressions.
sri« It is often necessary also to find the sum of ^n arith-
metical progression. This might be done by adding all the
terms together ; but as the addition vonld be Very tedious, when
the pi*ogression consisted of a great number of termsy a rule has
been devised, by which the sum may be more readily obtained.
372. We shall first consider a particular given progressioOf
such that the firat term = 2, the difference = 3, the last term
= £9, and the number of terms = 10 ;
123456789 10
e, 5, S, 11, 14, 17, SO, 23, 96, 29.
We see, in this progression, that the sum of the first and the
last term = 31 ; the sum of the second and the last but one
= 31 ; the sum of the third and the last but two s 31, and so
on$ and thence we conclude, that the sum of any two terms'
equally distant, the one from the first, and the other from the last
term, is always equal to the sum of the first and the last term.
373. The reasons of this may be easily traced. For, if we sup-
pose the first = a, the last = «, and the difference = d, the sum
of the first and the last term is = a -f ss ; and the second term
being = a -f d, and the last but one = ss — d, the sum of these
two terms is also rz a +%. Further, the third term being
a 4- 2 d, and the last but two = » — 2 d, it Is evident that these
two terms also, when added together, make a + %. The demon-
stration may be easily extended to all the rest
374. To determine, therefore, the sum of the progression pro-
posed, let us write the same progression term by term, invefted,
and add the corresponding terms together, as follows :
2 + 5 + 8 +11 + 14 + 17 + 20 + 23 + 26 + 29
29 + 26 + 23 + 20 + 17 + 14 + 11 + 8 + 5 + 2.
31 + 31 + 31 + 31 + 31 + 31 + 31 + 31 + 31 + 31
This series of equal terms is evidently equal to twice the sum
of the given progression ; now the number of these equal terms
is 10, as in the progression, and their sum, consequently, = 10
\
Chap. 4. Of Bairn and Proportion. ItST
X 31 = S10« So thatf since this svm is twice the sum of the
arithmetical progression, the sum required must be =s 155.
575. If we proceed in the same manner, with respect to any
arithmetical progression, the first term of which is = a, the last
= %9 and the number of terms = n ; writing under the given
progression the same progression inverted, and adding term to
term, we shall have a series of n terms, each of which will be
== a + » ; the sum of this series will consequently be = n (a + x),
and it will be twice the sum of the proposed arithmetical pro-
gression ; which therefore will be = ' ^ »
576. This result furnishes an easy method of finding the sum
of any arithmetical progression ; and may be reduced to the
following rule :
Jlftttt^Iy the sum of the first and the last term by the number rf
terms, and half the product will be the sum of the whole progres-
sion*
Or, which amounts to the same, multiply the sum of the first
and the last term by half the number of terms.
Or, multiply half Che sum of the first and the last term by the
whole number of terms* Each of these enunciations of the rule
will give the sum of the progression.
S7T. It may be proper to illustrate this rule by some exam*
pies.
First, let it be required to find the sum of the progression of
the natural numbers, 1, 2^ 3, &c. to 100. This will be> by the
first rule, = ^^ ^ ^^^ = 50 x 101 = 5050.
If it were required to tell how many strokes a clojck strikes
in twelve hours ; we must add together the numbers 1, 2, .3, as
12 vis
fkr as 12; now this sum is found immediately = — ^- — = 6 x
13 = 78. If we wished to know the sum of the same progres-
sion continued to 1000, we should find it to be 500500 ; and the
sum of this progression continued to 10000, would be 50005000.
378. JSnother question. A person buys a horse, on condition
that for the first nail he shall pay 5 halfpence, for the second 8,
for the third 1 1, and so on, always increasing 3 halfpence more
124 JBgetnu Sect S»
for each following> one ; tlie horse having $S nails, it is required
to tell how uiuch he will cost the purchaser?
In this question, it is required to find the sum of an arith-
metical progression, the first term of which is 5, the difference
=s 3, and the number of terms s 32. We must therefore be^in
by determining the last term ; we find it (by the rule in articles
365 and 370) = 5 + 31 X 3 = 98; After which the sum reqgir-
103 X 32
ed is easily found = — = lOB >c 16 ^ whence we conclude
that |he horse costs 1648 halfpence, or 3l. 8s. Sd.
379. Generally, let the firat term be = a, the difference = df
and the number of terms =; n ; and let it be required to find* by
means of the^e data, the sum of the whole progression. As the
last term must be = a -f (n — i) (i, the sum ot the first and Inst
will be = £ a -I- (n — 1) d. Multiplying this sum by the nutnber
of terms n^^e have 2na + n'(n-^l)(i; the sum required there-
fore will be = M a H — ^ .
This formula, if applied to the preceding example, or to a =,5,
i = 3, and n= 3^, gives 5 X 32 + ^^xSl x3 _ ^^^ ^ ^^^^ _
1648 ; the same sum that we obtained before*
380. If it be required to add together all the natural numbers
from 1 to n, we have, for finding this sum, the first term = !»
the last term =: n, and the number of terms = n ; wberelora the
Bum reqoimi is = "HLp = »^" + ^\
If we make n = 1766, the sum of all the numbera, from 1 to
1766, will be =: 883 x 1767 = 1560261.
381. Let the progression of uneven numbers be praposedf 1, 3, 5^
r, &c. continued to n terms, and let the sum of it be i*equired :
Here the first term is = 1, the difference = 2, the number of
terms = n; the last term will therefore be = 1 + (n— l) 2 =;
2n -^ 1, and consequently the sum required = nn.
The whole therefore consists in multi|ilying the number of
terms by itself. 8o that whatever number of terms of this pro*
gression ive add together^ the sum will be always a square^ namdy,
the square of the number of terms. This we shall eyenu^lify M
follows ;
Ciiap. 4. Cjf 1UiJ6m oskA Aioportum* 1 9X
Indices. l£S456f89 10&c.
Progress, 1, S, 5, 7» 9> llf 13^ 15, 17» 199 &c.
Sum, U 4^ 9, 16, 25, 36, 49, 64, 81, 100, &c.
382. Let the first term be=: 1, the difference =3, and the
Dumber of terms =; fi ; we shall have the progression 1, 4, 7,
10, &€• the last term of which will be 1 -l- ( n — 1) 3 = 3 n — 2 f
wherefore the sum of the first and the last term == 3 ft — ,1, and
.... - .. . . n (3 n— I) 3nn— ft
consequently, the sum of this progression = — ^ '= — - — ^
If we suppose n = 20, the sura will be = 10 x 59^ 590.
383. Again, let the first term = 1, the difference =s d, and the
number of terms = n ; then the last term will be=:l-f(ii — i)iL
Adding the first, we have 2 + (n — 1) d, and multiplying by the
number of terms, we have 2n + n(n— l)d; whence we deduce
the sum of the progression = n + 'Z •
We subjoin the following small table :
Ifd=l, thesumi8 = n+ iLl!LlJ) = ^!L±Jf
aJ 2
I d=:2, s=n+ — U — ^:=.nn
. - . Sii(ii — 1) 3nn — II
a = 3, := n H —^ ^ =
' ^2 2
J - . 4»(n — 1) ^
d = 4, =«H ^"5 =2nn — n
d = 5, =n+^^^'""^)=:^^^'^^^
d = 6, =«+^JL(]lZlL)=:3nn_8n
• w ^ , 7ii(n— 1) 7nn — 5it
2
d = 9, -„ , 9n(H-l) _ 9itn-r«
^ in . 10n(i>— 1) ^
4n
126 JI^Aira. fiflOt 5.
CHAPTER V.
()f Otomeirieal Bjoivo.
384. The geometrical ratio of two numbers is found by resolv-
ing the question^ ?unv many fifties is one of those numbers greater
than the other? This is done by dividing one by the others
and the quotient, therefore, expresses the ratio required.
385. We have here three things to consider; 1st, the first of
the two given numbers, which is called the antecedent ; 2dly, the
other number, which is called the co^iseqaent ; 3dly, the ratio
of the two numbers, or the quotient arising from the division of
the antecedent by the consequent. For example, if the relatioA
of the numbers 18 and 12 be required, 18 is the antecedent, 12
is ttie consequent, and the ratio will be 41 == i^ 9 whence we
see, that the antecedent contains the consequent once and a
half.
386. It is usual to represent geometrical relation by two
points, placed one above the other, between the antecedent and
the consequent. Thus a : 6 means the geometrical relation of
these t\io numbers, or the ratio of b to a.
We have already remarked, that this sign is employed to
represent division, and for this reason we make use of it here;
because, in order to know the ratio, we must divide a by (. The
relation, expressed by this sign, is read simply, a is to ft.
387. Relation therefore is expressed by a fraction^ whose
numerator is the antecedent, and whose denominator is the con*
sequent. Perapicuity requires that this fraction should be always
reduced to its lowest terms ; which is done, as we have already
shewn, by dividing both the numerator and denominator by
their greatest common divisor. Thus, the fraction \l becomes
|, by dividing both terms by 6.
388. So that relations only differ according as their ratios
are different ; and there are as many different kinds of geomet-
rical relations as we can conceive different ratios.
The first kind is undoubtedly that in which the ratio becomes
unity ; this case happens when the two numbers are equal, as
in ^1 : 3 ; 4 : 4 ; a : a ; the ratio is here 1, and for this reason
we call it the relation of equality.
Chap. 5. Of Batios and FrqporHon, 127
Next follow those relations in which the ratio is another whole
number ; in 4 : 2 the ratio is £> and is called daiMe ratio ; in
12 : 4 the ratio is 3, and is called tripU ratio ; in 24 : 6 the ratio
is 4, and is called quadruple ratio^ &c.
We may next consider those relations whose ratios are express
sed by fractions, as 12 : 9» wliere the ratio is 4 or 1-^ ; 18 : 27^
where the ratio is |» &c. We may also distinguish those relsi*
tions in which the consequent contains exactly twice^ tlurice, &x^«
the antecedent; such are the relations 6 : 12, 5 : 15, &c. the
ratio of which some call, subdupUf subtriple, &fi. ratios.
Further, we call that ratio raiioufU, which is an expressible
number; the antecedent and consequent being integers, as in
11 : 7, B : 15, &c« and we call that an irrational or surd ratio^
which can neither be exactly expressed by integers, nor by frac*
tions, as in vs": 8, 4 : V3~«
389. Let a be the antecedent, A the consequent, and d the ra-
tio, we know already that a and ft being given, we find d = ^*
If tlie consequent ft were given with the ratioi we should find
the antecedent a =: ft d, because ft d divided by ft gives d* Lastly,
when the antecedent a is given, and the ratio d, we find the
consequent ^ = -^ $ for, dividing the antecedent a by the conse-
quent -^f we obtain the quotient d, that is to say, the ratio.
390. Every relation a : ft remains the same, though we multL-
ply» or divide the antecedent and consequent by the same num-
ber, because the ratio is the same. Let d be the ratio of a : ft,
we have d =: t-; now the ratio of the relation n a : n ft b also
? = d, and that of the relation •- : — is likewise t- = d.
n n h
391. When a ratio has been reduced to its lowest terms, it is
easy to perceive and enunciate the relation. For, example, when
the ratio -r-has been reduced to the fraction ~, we say a : ft =
ft ^ •"
p : g, a : ft : : p : 9, which is ready a is to ft as p is to q. Thus,
the ratio of the i*elation 6 : 3 being f, or 2, we say 6 : 3 = 2 : 1.
128 JUgehra^ Sect. 9.
We have likewise 18 : 12 = 3 ; 2, and S4 : 1 8 = 4 : 3, and 30 : 45
^2:3, &c. But if the ratio cannot be abridged^ the relation
will not become more evident ; we do not siuiplirj the. relation
by s^ing 9 : 7 = 9 : 7*
392. On the other hand» we may sometimes chan|3^ the rela-
tion of two rery great numbers into one that shall be more
simple and evident* by reducing both to their lowest terms. For
example* we can say 28844 : 144tZ2 = 2 : 1 ; or,
10566 : 7041 = 3:2; or, 57600 : 25200 = 16:7.
393. In order, therefore, to express any relation in the clear^
est manner, it is necessary to reduce it to the smallest possible
numbers. This is easily done, by dividing the two terms of the
relaticm by their greatest common divisor. For example, to
reduce t)ie relation 57600': 25200 to that of 16 : 7, we have only
to perform the single operation of dividing the numbers 576 and
252 by 36, which is their greatest common divisor.
3lk. It is important, therefore, to know how to find the great-
est common divisor of two given numbers ; but this requires a
rule* which we shall explain in the following chapter*
CHAPTER VI.
Of the greatest Common Divisor of two gvoen numbers.
395. Thebe are some numbers which have no other common
divisor than unity, and when the numerator and denominator
of a fraction are of this nature, it cannot be reduced to a more
convenient form. The two numbers 48 and 35, for example^
have no common divisor* though each has its own divisors.
For this reason we cannot express the relation' 48 : 35 more
simply, because the division of two numbers by 1 does not
diminish them.
396. But when the two numbers have a common divisor* it is
found by the following rule :
Divide the greater of the two numbers by the less ; nextf divide
the preceding divisor by the remainder; what remains in this
second division will afterwards become a divisor for a third divis-
ion, in which the remainder of the preceding division will be the
Chap» & Cf RaiioM and Fropartiaa. 1 S9
dividend. We must amtimie this operation^ tiU ube arrive at a
division thai leaves no remainder ; the divisor of this division^ and
conseqnenUy the last dixisoTf will be the greatest common divisor oj ^
the two given nunibers*
See this operation for the two numbers 576 and 252.
252) 576 (2
504
, 7i) 252 (3
216
36) 72 (2
72
0,
So that* in this instancet the greatest conmon divisor is 36.
397. It will be proper to illustrate this rule by some other
examples. Let the greatest common divisor of the numbers
504 and 312 be required.
312) 504 (1
312
192)312(1
192
120) 192 (1
120
72) 120 (1
72
48)72(1
48
24) 48 (2
48
Srp that 24 is the greatest common divisor^ and consequently
the i^elation 504 : 312 is reduced to the form 21 : 13« '
S^, Let the relation 625 : 529 be given, and the greatest
comnbon divisor of these two numbers be required.
JBvB. JUg. 17
Ki
ISO JIgelmu .Beet d«
529) 625 (i
529
96) 529 (5
480
49) 96 (1
49
47) 49 (1
47
2) 47 (25
46
1) 2 (2
2
0.
^Wherefore 1 is^ in this case, the greatest common divisor^
and consequently we cannot express the relation 625 : 529 by
less numbers, nor reduce it to less terms.
399. It roaj be proper* in this place, to give a demonstration
of the rule. In order to this, let a be the greater and 6 the less
of the given numbers ; and \etd be one of their common divisors ;
it is evident that a and b being dirisible by d, we may also
divide the quantities a-^b, a — 2 (, a — S fr, and, in general^
U'^^nb by d.
400. The converse is no less tnie ; that is to say, if the num-
bers b and a*— n 5 are divisible by d, the number a will also be
divisible by d. For nb being divisible by d, we could not divide
a — n ft by d, if a were not also divisible by d.
401. We observe further, that If d be the greatest common
divisor of two numbers, b and a — n 6, it will also be the great-
est common divisor of the two numbers a and A. Since, if a
greater common divisor could be found than d, for thtne num-
bers, a and ft* that number would also be a common divisor t of 6
and a — nb; and, consequently, d would not be the greatest
Qommon divisor of these two numbers. Now we have supnosed
d the greatest divisor common to b and a — nb; whet*ef^re d
must abo be the greatest common divisor of a and i. .'
<?
Chap. S» Of J&ittM and Frqporfiaa. 131
402. These three things being laid down^ let us divide^
according to the rule, the greater number a by the less h ;
and let us suppose the quotient = n ; the remainder will be
a i— fi (9 which must be less than ft. Now this remainder a^^nh
having the same greatest common divisor with ft, as the given
numbers a and 6, we have only to repeat the division, dividing
the preceding divisor i by the remainder a— ->fifr; the new
remainder, which we obtain, will still have, with the preceding
divisor, the same greatest common divisor, and so on*
403. We proceed in the same manner, till we arrive at a
division without a remainder ; that is, in which the remainder
is nothing. Let p be the last divisor, contained exactly a cer-
tain number of times in its dividend ; this dividend will there-
fore be divisible by p, and will have the form mp ; so that the
numbers p, and mf^ are both divisible hyp; and it is certain,
that they have no greater common divisor, because no number
can actually be divided by a number greater than itself. Con-
sequently, this last divisor is also the greatest common divisor
of the given numbers a and 6, and the rule, wbich we laid down,
is demonstrated.
404. We may give another example of the same rule, requir-
ing the greatest common divisor of the numbers 1728 and 2304.
The operation is as follows :
ir28) 2304 (1
1728
576) 1728 (S
1728
*^"— ■■•
0.
From this it follows, that 576 is the greatest common divisor,
and that the relation 1728 : 2304 is reduced to 3 ; 4 ^ that is to
eay^ 1728 is to 2304 the same as 3 is to 4»
1st Jigdnu Ca(tS»
GHAPTEB VIL
Of OeBmetfical Proportions,
405. Tiro geometrical relations are equal* when their ration
are equal. This equality of two relations ia raUed a geomehieal
proportion ; and we write for example, a'Jbszc^.dfUraibiicidp
to indicate that the relation t^zbis equal to the relation c : d ;
but this is more simply expressed by s^yingt a is to ^ a«» c to 4*
The fdKiwiag ia such a proportHMi» 8 : 4 =? 12 : 6 ; for the ratio
of the relation 8 : 4 is |» and thi9 is also the ralio of the rela*
tion 12:6,
406. So that atbzsczd being a geometrical proportion, the
a
ratio must be the same on both sides, and -r = -^ 3 ^"^> recipro-
cally, if the fractions -^ and -^ are equal, we have aibiicid.
407. A geometrical proportion consists therefore of four terms^
such, that the firsts divided by the second^ gives the same quo*
tient as the third divided by the fourth. Hence we deduce ai|
important property, common to all geoaM^trical pro|Kirti6n»
which is, that the product of the first and the last term is tdi^aye
equal to the product of the second and third ; or, more simply, that
the product of the extremes is equal to the product of the means.
408. In order to demonstrate this property, let us tal^e the
a c
geometrical proportion a : 5 = c : dj. so that -r^-^* If we mul-
he •
tiply both these fractions by (, we obtain a sb <^, and multiply-
ing both sides fui*ther by d, we have ad:=ibc. Now a d is the
product of the extreme terms, ft c is that of the means, and theso
two pi*oducts are found to be equak
409. Reciprocally 9 ^ the Jour numbers a, b, c, d, are suckj that
iheproditct of the two extremes a and d is equal to the product of
the two means h and c, we are certain thtit tiieyform a geometric
eal proportion. For since adssbCf we have only to divide both
sides by ft d, which gives ^ Ca^ b2* ^^ T^ "d* ^^^ consequent-
ly a:ft=s:c:d«
Chi^. 7. Of Itatias and Frcporiiam. 1$$
410. Tlefmr terms (f a geowuirieel jnvpwHtmfOB VLih
may be transposed in dxjjfertHt waySf witkout destrajfhig the pnh
portion. For the nde being alwayst that the product of the ex^
tremes is equaltotheproductofthefneanSfOrAiz=hc,wemaysay^
!•*• b : a = d : c ; fi^^* a : c s b : d ; 5*^* d ; b = c : a ;
4«wy* d : c = b ; a.
4 U Besides thesQ four geometrical proportions* we may de-
dare some others from the same proportioiiy a ibz^c : d. We
may «ajf, the first term^ plus the second^ is to the firsts as the third
+ the fourth is the third ; that i8»a-f.b:a = c+d; cc
We may furtlier say ; the fir si — the second is to the first as the
Unrd — fhejourth is to the thirds or a — b ; a:=c — d : c»
For, if we take the product of the extremes and the imeans,
we have ac — bez=iac — ad, which evidently leads to the equaU
ity a d = fr c.
Lastly, it is easy to demonstrate, that a + b :b=:c + d : d;
and that a* — 6 : J = c — did.
4)2. All the proportions which we have deduced from a : 6 =
e : df may be representecU generally, as follows :
ma-^nbipa + qbs:mc + n dipe + qd.
For the product of the extreme tefms ismp a c+n p b c+m qad
+n qbd ; ^hich» sincen d = 6 c, becomes ni pac+npbc + mqbc
+ nqbd. Further, the product of the mean terms is m p a c -f.
mqbe + npad + nqbdi or* since a d=: ft c, it is mp ac-(-m 9ft c
-f-tij^&c-fngfrd; so that the two products are equal.
413* It is evident, therefore, that a geometrical proportion
being gi%en, for example, 6 : 3 = 10 : 5, an infinite number of
ethers may be deduced from it We shall give only a few :
S ; 6 = 5 : 10; 6 : 10 = 5 5; 9 : 6= 15 : 10;
, 5:3 = 5: 5; 9 : 15 = 5 : 5 ; 9 : 5 = 15 : 5.
414. Since, in every geometrical proportion, the product of
the extremes is equal to the product of tli6 means, we may,
when the three first terms are known, find the fourth from them.
Let the three first terms be 24 : 15 =40 to . • . . as the product
of the meitns is here 600, the fourth term multiplied by the firsts
that is by 24, roust also make 600 ; consequently, by dividing
600 by 24, the quotient 25 will be the fourth term required, and
the whole proportion will be 24 : 15 = 40 : 25. In general^
134 JBgAriU Sect 3»
thereforef if the three first terms aoea :issc:.«.. weputd
for the unknown fourth letter ; and since a d = i c^ we diyide
be
both sides hy a, and have d = — . So that the fourOi term is
^ — f y and is found hy muUiplying the second term by the thirdf and
a
dividing that product by the first temm
415. This is the foundation of the celebrated Rule(f Three in
arithmetic ; for what is required in that rule 7 We suppose three
numbers given, and seek a fourth, which may be in geometrical
proportion ; so that tlie first may be to the second, as the third
is to the fourth.
416. Some particular circumstances deserve attention here.
First, if in two proportions the first and the third terms are the
samCf as in a : 6 = c : d, and a :f=c : gf I say that the two
secmd and the two fourth terms will also be in geometrical propor^
tionf and that b:d=^fig. For, the first proportion being
transformed into this, a : c=5 : d, and the second intp thist
a : c=:f:gf it follows that the relations b ; d and/: g are equals
since each of them is equal to the relation a : c For example,
if 5 : 100 = £ : 40, and 5 : 15 =:2 : 6, we must have 100 : 40
= 15 : 6.
417. But if the two proportions are such, that the mean terms
are tlie same in both^ I say that the first terms will be in an
inverse proportion to the fourth terms. . That is to say, if a : 6
= e: df and/: b=ic: gt it follows that a if^g : d. Let the
proportions be, for example, S4 : 8 = 9 : 3, and 6 : 8 = 9 : IS^
we have 24 : 6 = IS : 3. The reason is evident ; the first pro-
portion gives adzi^bc; the second eiyesfg = bc; therefore^
a d =fg9 and a :f=g idforaig: :fi d.
418. Two proportions being given, we may always pi'oduco
a new one, by separately multiplying the first term of the one
by the first term of the other, the second by the second, and so
on, with respect to the other terms. Thus, the proportions a : b
=ic: d and e :/= g : h will furnish this, a e : 6/= eg idh. For
the first giving adzzibc^ and the second giving e h ^fg$ we have
also adeh:=ib cfg. Mow a d e A is the product of the extremes,
and b cfg is the product of the means in the new proportion ; so
that the two products being equal, the proportion is truer
Chap* 8. Of Batm and Praporliotu 135
419. Let the two proportions be^ for example, 6 : 4 := 15 : 10
and 9 : 12 = 15 : 20, their combination will give the proportion
6x9:4x 12=15X15 : 10X20,
or 54 : 4S = 225 : 200^
or 9 : 8 = 9 : 8.
420. We shall observe lastly, that if two products are eqnaU
a d = ft c, we may reciprocally convert this equality into a geo-
metrical proportion ; for we shall always have one of the factors
of the first product, in the same proportion to one of the factors
of the second product, as the other factor of the second product
is to the other factor of the first product ; that is, in the present
case, a : c=: ft : d, or a : b =E c : d. Let 3 x 8 =4 x 6, and we
may form from it this proportion, 8 : 4 = 6 : 3, or this^ 3:4 =
6 : 8, Likewise, if 3 x 5 = 1 X 15, we shall have
3 : 15 =1 : 5, or 5 : 1 = 15 : 3, or 3 : 1 = 15 : 5.
CHAPTER VIIL
Mservaiions on the Ruks of Proportion and their utitity.
421. This theory is so useful in the occurrences of common
life, that scarcely any person can do without it. There is always
a proportion between prices and commodities ; and when differ-
ent kinds of money are the subject of exchange, the whole con-
sists in determining their mutual relations. The examples^
furnished by these reflections, will be very proper for illustrating
the principles of proportion, and shewing their utility by the
application of them.
4^2. If we wished to know, for example, the relation between
two kinds of money ; suppose an old louis d'or and a ducat ; we
must first know the value of those pieces, when compared to
others of the same kind. Thus, an old louis being, at Berlin^
worth 5 rix dollars* and 8 drachms, and a ducat being worth
3 rix dollars, we may reduce these two values to one denomina-
tion ; either to rix dollars, which gives the proportion 1 L : 1 D
• The rix dollar of Gennany is yalued at 92 cents 6 mills, and « drachm it
one twenty-fourth part of a rix dollar. *
136 Mgebm. Seci
s=54R:SR» or=16:9; or to drachmSf in which case w«
have 1 L : 1 O = 128 : 72 = 16 : 9. These proportions trU
dently give the true relation of the old louis to the ducat ; for
the equality of the products of the extremes and the means givesy
in bothy 9 louis = 16 ducats ; aiid» by means uC thi^ comparisoiiy
we may change any sum of old louis into ducats, and vice versa.
Suppose it were required to tell how many ducats there are in
1000 old iouisy we have this rule of three. If 9 louis are equal
to 16 ducats* what are 1000 louis equal to ! The auswer will h%
1777; ducats.
It, on the cnntrary, it were required to find how many oM
louis d'or there are in 1000 du<*ats» we have the following pro*
portion. If 16 ducats are equal to 9 louis; what are 1000
ducats equal to ? ^nswer^ 562| old louis d'or*
4St3« Here, (at Petersburg,) the value of the ducat yarieSf
and depends on the course of exchange. This course determines
the value of the ruble in stivers^ or Dutch half-pence^ 105 of
which make a ducat.
So that when the exchange is at 45 stivers, we have this pro-
portion, 1 ruble : 1 ducat = 45 : 105 =3:7; and hence this
equality, 7 rubies = 3 ducats.
By this we shall find the value of a ducat in rubles ; for 3
ducats : 7 rubles = 1 ducat : JinswcTf £^ rubles.
If the exchange were at 50 stivers, we should have this pro-
portion, 1 ruble : 1 ducat = 50 : 105= 10 : 21, which would
give 21 rubles = 10 ducats ; and we should have 1 ducat = Q^^
rubles. Lastly, when the exchange is at 44 stivers, we have I
ruble : 1 ducat = 44 : 105, and consequently 1 ducat s 2^^
rubles = 2 rubles SS-f^ copecks.*
424. It follows from this, that we may also compare different
kinds of money, which we have frequently occasion to do in billi
of exchange. Suppose, for example, that a person of this place
has 1000 rubles to be paid to him at Berlin, and that he wishes
to known the value of this sum in ducats at Berlin.
The exchange is here at 47|, that is to say, one ruble makes
47| stivers* In Holland,^ 20 stivers make a florin ; 2| Dutch
florins make a Dutch dollar Further, the exchange of Holland
*^ A copeck 18 Y^ part of a ruble, as is eaiily dedutcd from the above.
Chap, s; Of Ratios imd'-frtfuniian. 1S7
xvith Berlin is at 142. that is to say, for 100 Datch dollars, 142
doDars are paid at Berlifi. Lastly^ the ducat is worth 3 dollars
at Berlin.
425. To resolve the questions proposed, let us proceed step
by step. Beginning; therefore wkb the stivers, since 1 ruble =?
47*1 stivers, or 2 rubles = 95 stivers, we shall have 2 rubles : 95
stivers = 1000 : • . . . Mswer^ 47500 stivers. If we go fur-
ther and say 20 stivers : 1 florin = 47500 stivers: .... we sliall
have 2375 florins. Further, 2| florins -^ 1 Dutch dollar, or 5
florins = 2 Dutch dollars ; we shall therefore have 5 florins : 2
Dutch dollars ^ 2S7i^ florins :•••• Answer^ 950 Dfttch dollars.
Then taking the dollars of Berlin, according to the exchange
at 142, we shall have 100 Dutch dollars : 142 d<illara = 950 :
the fourth term, I S49 dollars of Berlin. Let us, lastly, pass
to the ducats, and say S dollars : 1 ducat = 1349 dollars : . • . •
Answer, 449| ducats.
426. In order to render these calcolatvops st|ll more complete^
let us suppose that the Berlin banker refuses, under some pre-
text or other, to pay this sum, and to accept the bill of exchange
without five per cent, discount ; that is, paying only 100 instead
of 105. In that case, we must make use of the following pro-
portion ; 105 : 100 = 449f : a fourth term, which is 428|^
ducats.
427. We have shewn that six operations are necessary, in
making use of the Rule of Three ; but we ran greatly abridge
those calculations, by a rule, which is called the Ride of Reduc^
turn. To explain this rule, we shall first consider the two
antecedents of eac h of the six operations.
I. 2 rubles : 95 stivers.
II. 20 stivers : 1 Dutch flor.
III. 5 Dutch flor. : 2 Dutch doll.
IV. 100 Dutch doll. : 142 dollars.
y. 3 dollars : 1 Ducat.
YI. 105 ducats : 100 ducats.
If we now look over the preceding calculations, we shall ob-
serve, that we have always multiplied the given sum by the
second terms, and that vse hjive divided the products by the
first ; it is evident therefore^ {hat wjg shall arrive at the same
Etd. JUg. 16
Ids
Sect3«
resultsy by multiplying, at once^ the sum proposed by the pro-
duct of all the second terms, and dividing by the product of all
the first terms. Or, which amounts to the same things that we
have only to make the following proportion ; as the product of
all the first terms is to the product of all the second terms, so is
tlie .fi^iveo number of rubles to the number of ducats payable at
Berlin.
428. This calculation is abridged still more, when amongst
the first terms some are found that have common divisors with
8f)me of the second terms ; for, in this case, we destroy those
terms, and dubstitute the quotient arising from the division by
that common divisor. The preceding example will^ in this
manner^ assume the following form.*
Rubles ^ : 19 jflf stiv. 1000 rubles.
1 Dutch flor.
^ Dutch dollars.
14£ dollars.
1 ducat*
^f/Vft ducats.
lWi21.
6Sf#
£698= 1051^:-*
7) 26980.
9) 3854 (2
428 (2. Msweff 428 Jl ducats.
429. The method, which must be observed, in using the rule
of reduction, is this ; we begin with the kind of money in ques^
tiuin, and compare it with another, which is to begin the next
relation, in which we compare this second kind with a thirds
and so on. Each relation, therefore, begins with the same kind^
as the preceding relation ended with. This operation is con-
tinued, till we arrive at the kind of money which the answer
requires ; and, at the end, we reckon the fractional remainders.
• Divide the 1st and Mr by 2»^tbe Ji^ and 12th by 20» the 5th and 13th
(which is now 5) by 5, also the #3 And 11th by 5»
V
430. Other examples are added to facOitate the practice of
ttkis calculatioti.
If ducats gain at Hamburg 1 per cent on two dollars banco ;
that is to say, if 50 ducats are worth* not luo, but 101 dollars
banco ; and if the exchange between Hamburg and Konigs-
berg is 119 drachms of Poland ; that is, if 1 dollar banco gives
119 Polish drachmsy how many Polish florins will 1000 ducats
give?
30 Polish drachms make 1 Polish florin.
Ducat 1 : 9^ doll. B^ 1000 due.
JljI^fSO : 101 doll. B'*. ^
1 : 119 Pol. dr.
SO : 1 PoLflor.
••mmmmmami^^»
15^ : 12019 = lOfi/tf Aiic. :
3) 120190.
5) 40063 (1.
80 1£ (3. Jlfiswer^ 8012| P. tL
431. We Aiay abridge a little further, by writing the number,
which forms the third term, above the second row ; for then the
product of the second row, divided by the product of the first
row , will give the answer sought.
^stion. Ducats of Amsterdam are brought to liCipsick^
having in the former city the value of 5 flor. 4 stivers current }
that is to say, 1 ducat is worth 104 stivera, and 5 ducats are
worth 26 Dutch florins. If, therefore, the agio of the bank* at
Amsterdam i^ 5 per cent., that is, if 105 currency afre equal to
100 banco, and if the exchange from Leipsick to Amsterdam,
in bank money, is 33^ per cent that is, if for 100 dollars wo
pay at Leipsick 133^ dollars $ lastly, 2 Dutch dollars making
5 Dutch florins; it is required to find how many dollars we
must pay at- Leipsick, according to these exchanges, for 1000
ducats?
* The difference of vdoe between bank money and current money.
140
^i'
AV
Ducats
f^ /i^|^|Kducatfl.
26 flor. Dutch curr.
49)1^9 ^/flor. Dutch banco.
533 doll, of Leipsick*
/B dolL banco.
21 : 3) 05432 (L
^M
7) 18477(4.
/^ 2639.
JkMrwtr^ 2639ff dollars, or 2639 dollars and 15 dracbmi.
CHAPTER IX.
Of Compound Relations*
432* CoMFOTTND RELATIONS are obtained, by multiplying the
terms of two or more relations, the antecedents by the antece-
dents, and the consequents by the consequents; «e say then,
that the relation between those two products is compounded of
the relations given.
Thus, the relations aib, c: i, ex J, give the compound rela-
tion ace : 6d/.*
433i^ A relation continuing always the same, when we divide
both its tterms by the same number, in order to abridge it, we
may greatly facilitate the abi»ve composition by comparing the
antecedents and the consequents, for the purpose of making
such reductions as we performed in tlie last chapter.
For exaiople, we find the compound relation of the following
given rdations^ihus ;
* Each of these three ratios is said to be one of the r^te of the compound
ratio.
Chap* 9. Of Batioi ami ProporHon. '* I'^i
BdaiiMS giveiu
12 : 25, 28 : SS, and 55 : 56.
2 : 5.
So that 2 : 5 is the ^ompound relation required.
434. The same iteration is to be performed, when it is re-
quired to calculate g^dlierally by letters ; and the most remark-
able case is that, in which each antecedent is equal to the
Gonsequeat of the preceding relation. If the given relations are
a: b
b : c
c : d
d : e
e : a
the compound relation is 1 ; 1 •
435. The utility of these principles will be perceived, when
it is observed, tiiat the relation between two square fields is
compounded of the relations of the lengths and the breadths.
Let the two fields, fur example, be A and B ; let A liave 500
feet in length by 60 feet in breadth, and let the length of B be
560 feet, and its breadth 100 feet ; the relation of the lengths
will be 500 : 360, and that of the breadths 60 : 100. So that
we have
W I >^/-
5 : 6
Wherefore the field A is Co the field B, as 5 to 6,
436. Jinother example. Let the field* A be 720 feet long, 88
feet broad ; and let the field B be 660 feet long, and 90 feet
broad ; the relations will be compounded in the following man-
ner.
Belation of the lengths, ^^ 8 : 15 ^^^fi
Eelation of the breadths, S% ;^, 2 : j^
ff^i,^'.
Relation of the fields A and B, 16 15.
14itr JS^a. Sect 9.
43r. FttrHiert if it be required to compare two chambers with
respect to the spactf or contents, we observe tW that relation
is compounded of three relations; namely, of that of the
lengths^ that of the breadths, and that of the heights. Let there
be, for example, the chamber A, whose length = 36 feet, breadth
= 16 feet, and height s 14 feet, and the chamber B, whose
length = 4£ feet, breadth = 24 feet^ and height s 10 feet j we
shall have these three relations ;
For the length /jfCji :
For the breadth ^ /> 2 :
For the height 1^, £ :
4 : 5
So that the contents of the chamber A ; contents of the cham-
ber B, as 4 : 5.
436. When the relations, which we compound in this manner,
are equal, there result multiplicate relations. Namely, two
equal relations give a duplicate ratio or ratio of tht squares;
three equal relations produce the triplicate ratio or ratio of the
cubeSf and so on, for example, the relations a : b and a : b give
the compound relation aaibb; Mherefore we say, that the
squares are in the duplicate ratio of their roqits. And the ratio
a : 5 multiplied thrice, giving the ratio a' : fr', we say that the
cubes are in the triplicate ratio of their roots,
439* Greometry teaches, that two circular spaces are in the
duplicate relation of their diameters ; this means, that they are
to each other as the sqAares of their diameters*
Let A be a circular space having the diameter = 45 feet, and
B another circular space, whose diameter = 30 feet ; the first
space wQI be to the second, as 45 x 45 to 30 x 30 ; or, com-
pounding these two equal relations,
fii.p^ 3 : 2, /{, %ii.
* 9:4.
'Wherefore the two areas^ are to each other as 9 to 4.
440. It is also demonstrated, that the solid contents of spheres
are in the ratio of the cubes of the diameters. Thii% the diam^-
Chap. 9» Cf BaHas and Proporfiam
ter of a j^obe Af being 1 foot, and tlMp diameter of a globe
being % feet, the solid contentB of A will be to those of B^ as '
1^ : £' ; or, as 1 to 8.
If therefore, the spheres are formed of the same substancet
the sphere B will weigh 8 times as much as the sphere A.
441. It is evident, that we may, in this manner, find tha
weight of cannon balls, their diameters, and the weight of onot
being given. For example, let there be the ball A, whosd^
diameter = 2 inches, and weight = 5 pounds ; and, if the
weight of another ball be required, whose diameter is 8 inches^
we have this proportion, 2^ : 8' =s 5 to the fourth term, 320
pounds, which gives the weight of the ball B. For another ball
C, whose diameter =15 inches^ we should have,
2« : 15« = 5 : . . . Jkmoer, 2109| lb.
o c
442. When the ratio of two fractions^ ^H' T* ^ requir-
ed, we may always express it in integer numbers*; for we have
only to multiply the fractions by b (f, in order to obtain the
ratio adibCf which is equal to the other ; from which results the
proportion -r : ^ = a d : ft c If, therefore, ad and h c have com-
mon divisors, th^ ratio may be reduced to less terms. Thus,
If : If = 15 X 36 : 24 X 25 = 9 : 10.
443. If we wished to know the ratio of the fractions — and -r »
a b
.it is evident, that we should have ~ : -^ = 5 : a ; which is ex-
pressed by saying, that two ftadtUmB^ 'which have unity for their
numeratorf are in the reciprocal^ or infoerse ratio oj thdr denomi'
nators^ Hie same may be said of two fractions, which hare any .
c e
common numerator; for ^ : -j-szbia* But if two fractions have
a h
their denominators equal, as — : —, they are in the direct ratio of
{he numerators ; namely, as a : fr. Thus* | : -/^ = -f^ : ^^^ = 6 : 3
=3 2 ; 1, and V : V = ^^ * ^^^ or, = 2:3.
444. It is observed^ that in the free descent of bodies, a body
144 Mgebra. Sect S.
falls 16* feet in a second^ that in two seconds of time it falls
64 feety and that in tliree seconds it falls 144 feet ; hence it is
concluded, that the heights are to one another as the squares
of the times ; and that, reciprocally, the times are in the sub-
duplicate ratio of the heights^ or as the square roots of the
heights.
If, therefore, it be required te find bow long a stone must
taka to fall from the height of 2S04 feet ; we have 1 6 : SS04 = I
to the square of the time sought. So that the square of the lime
sought is 144 ^ and, consequently, the time required is IS seconds.
445. It is required to find how far, or through what height^
a stone will pass, by descending for the space of an hour; that
is, 360Q seconds. We say, ther^ore, as tlie squares of the timesy
that is, I * : S600> ; so is the given height db 16 feet, to the
height required.
1 : 12960000 = 16 : . . . . 207360000 height required.
16 '
77760000
1296
207360000
If wc now reckon 19200 feet for a league, we shall find this
height to be 10800 ; and, consequently, nearly four times greater
than the diameter of the earth.
446. It is the same with regard to the price of precious stones,
which are not sold in the proportion of their weight; every
body knows that their prices follow a much greater ratio. The
rule for diamonds is, thatthe price is in the duplicate ratio of
the weight, that is to say, the ratio of the prices is equal to the
square of the ratio of the weights. The weight of diamonds is
expressed in carats, and a carat is equivalent to 4 grains ; if,
therefore, a diamond of one carat is worth 10 livrcs, a diamond
of 100 carats will be worth as many times 10 livres, as the
square of 100 contains 1 ; so that we shall have, according to
the rule of three,
* 15 is used in the original, as expressing the dlescent in Paris fceti It Is
here altered to English feet
Gbap. 9. BaHos and ProporUotu 149
1» : 100* =» lOlivres,
or 1 : 10000 = 10 : • • . • Jinswer, 100000 livres.
There is a diamond in Portugal, which weighs 1680 carats ; its
price will be found, therefore, by making
!• : 1680» = lOli? :.•. .or
1 : 28£2400 = 10 : £8224000 IiF.
447* The posts or mode of travelling in France furnish exam-
pies of compound ratios, as the price is according to the com«
pound ratio of the number of horses, and the number of leagues^
or posts* For example, one hoi^se costing 20 sous per post, it
is required to find how much is to be paid for 28 horses and 4|
posts*
We write first the ratio of horses, 1 : 28,
Under this ratio we put that of the stages or posts, 2 %
And, compounding the two ratios, we have 2 : 252^
Or, 1 : 126 = 1 livre to 126 francs or 42 crowns.
Mather q^itstum. If I pay a ducat for eight horses, for S
German miles, how much must I pay for thirty horses for four
miles 7 Thexalculatiqn is^ as follows :
^jl ^^f'%
V
1 : 5, = 1 ducat : the 4th term, which will be
5 ducats.
448. The same composition occurs, when workmen are to ba
paid, since those payments generally follow the ratio compound-
ed of the number of workmen, and that of the days which they
have been employed.
If, for example, 25 sous per day be given to one mason, and
it is required to find what must be paid to 24 masons who have
worked for 50 days \ we state this calculation ;
1 : 24
1 : 50
1 : 1200 = 25 : • . . . 1500 francs.
25
20)80000(1500.
EmL Alg. 19
146 JUgdnxL Sect. S.
As, in such examples, five things are giveiif the rule, which
serves to resolve them, is sometimes called, in books of arith-
metic, The Rule of Five,
CHAPTER X.
Of Oetmetrical ProgressUms,
449. A 8ERIB8 of numbers, which are always becoming a
certain number of times greater or less, is called a geomdrieal
progressiofif because each term is constantly to the following one in
the same geometrical ratio. And the number which expresses
how many times each term is greater than the preceding, is
called the exponent Thus, when the first term is 1 and the
exponent = 2, the geometrical progression becomes.
Terms 123 4 5 6^7 89 &C.
Prog. 1, 2, 4, 8, 16, 32, 64, 128, 256, &c.
the numbers 1, 2, 3, &c. always marking the place Which each
term hoMs in the ))rogression«
450. If we suppose, in general, the first term = a, and the
exponent = b, we liave the foUowing geomdrieal progression ;
1, ^ 3, 4, 5, 6, 7, 8 « • • • n
Progia, ah ab^, a t«, a 6*, a 6*, ab^, ab'' . . ..a 6"~*.
So that, when this progression consists of n terms, the last
term is = a b^K We must remark here, that if the exponent b
be greater than unity, the terms increase continnally ; if the ex*
ponent fc = 1, tlie terms are all equal ; lastly, if the exponent b
be less than 1, or a fraction, the terms continually decrease.
Thus, when a = 1 and ft =: j., we have this geometrfeal progres-
sion ;
11111 1 1 I. Aep
451. Here therefore we have to consider ;
I. The first term, which we have called a.
II. The exponent, which we call 5.
III. The number of terms, which we have expressed by «•
IV. The last term, which we have found = a 6"""\
So that, when the three first of these are given, the last term is
Cha|i. 10. BaUM and Proporfioii. 147
faaad» bj nraltiplytng the n — 1 power of &» or 6"~S by the first
term (u «
Iff therefore^ the 50th term of the geometrical progression
If 2» 4t 8f &c. were required* we should have a = 1, & = 2t and
nzsSO; consequently the 50th term =s 2^ * • * Now d * being s 5 12 ;
£^<> will be = 10£4« Wherefore the square of 2i«» or 2'«» as
1048576> and the square ^f this number, or 1099511627776 s
£«•. Multiplying therefore this value of 2^<^ by 2*, or by 512,
we hare 2«* equal to 562949953421312.
452* One of the principal questions, which occurs on this
subject, is to find the sum of all the terms of a geometrical pro-
gression ; we shall therefore explain the method of doing this*
Let there be given, first, the following progression, consisting of
ten terms;
1, 2, 4, 8, 16, 32, 64, 126, 256, 512,
the sum of which we shall represent by s, so that s = 1 -f. 2 -f
4 + 8 4- 16 + 32 + 64 -f 128 + 256 + 512; doubling both sides,
we shall have 2 s = 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 +
512 + 1024. Subtracting from this the progressiori represented
by s, there remains s =: 1024 — 1 = 1023 ; wherefore the sum
required is 1023.
453. Suppose now, in the same progression, that the number
of terms is undetermined and = n, so that the sum in question,
or s, = 1 + 2 + 2» + 2» + 2* . • . . 2«-». If wo multiply by 2,
we have 2 s = 2 + 2* +2' +2^ .... 2*, and subtracting from
this equation the preceding one, we have s == 2" — 1. We see,
therefore, that the sum required is found, by multiplying the last
term, 2""^, by the exponent 2, in order to have 2", antf subtract-
ing unity from that product.
454. This is made still more evident by the following exam-
ples, in which we substitute successively, for n, the numbers 1, 2,
3,4, &C.
1 = 1; 1+2 = 3; 1+2+4 = 7; 1+2+4 + 8=15;
1+2 + 4+8 + 16 = 31; 1 +2 +4 + 8 + 16 + 32 = 63,&C.
455. On this subject the follomng question is generally pro-
posed. A man oflTers to sell his horse by the nails in his shoes,
which are in number 32 ; he demands 1 Hard for the first naif.
14S ^ Mgebra. Sect S.
2 for the second, 4 for the third, 8 for tiie fourth, and so on, do-
mandiiig for eacli nail twice the price of the preceding. It is
required to find what would be the price of the horse?
This question is evidently reduced to finding the sum of all
the terms of the geometrical progression 1, 2, 4, 8, 16, &c. con-
tinued to the 32d term. Now this last term is 2' ^ ^ and, as we
have already found £*^ = 1048576, and 2^<^ = 1024, we shall
haveS** x 2*« = 2»<> equal to 1073741824 5 and multiplying
again by 2, the last term 2^^.= 2147483648 ; doubling there-
fore this number, and subtracing unity from the product, the
sum required becomes 4294967^295 liards. These Kards make
107374 182d| sous, and dividing by 20, we have 53687091 livres,
3 sons, 9 denici*s for the sum required.
456. Let the exponent now be a 3, and let it be required to
find the sum of the geometrical progression 1, 3, 9, 27, 81, 243,
729, consisting of 7 terms. Suppose it s: 9, so that
S = 1 +3 + 9 + 27 + 81 + 243 + 729 ;
we shall then have, multiplying by 3,
3 5 = 3 + 9 + 27 + 81 + 243 + 729 + 2187 ;
and subtracing the preceding series, we have 2 5 = 2187 -* 1 =:
2186. So that the double of the sum is 2186, and consequently
the sum required = 1093.
457. In the same progression, let the number of terms =s n, and
the sum = « ; so that 5 =j= i + 3 + 3* + 3* +3* + . . . . 3»-*.
If we multiply by 3, we have 3 s = 3 + 3* + 3' + 3f + . . . . 3 *.
Subtracting from this the value of Sf as all the terms of it,
except the first, destroy all the terms of the value of 3 5, except
3*«^l
the last, we shall have 2«=:3" — 1: therefore % == — ---. So
2
that the sum required is found by multiplying the last term by
3, subtracting 1 from the product, and dividing the remainder
by 3* This will appear, also, from the following examples ;
1 = 1; l+^ = «Ji|=l=4, 1+3 + 9 = 1^== IS 5
3v27— •!
1+3 + 9+27= -~ — - es 40j 1 + S + 9 + 27 + 81 =
^_2<!l=l^i2i.
o
Chap. 10« Uaiios and Proportion. 149
458. Let us now suppose^ geiierally» the first term = a» the
exponent = 6f the number of terms = fh and their sum = St so
that
8=:atlBah + ab^ +ab^ +ab^ +. • . . al/^K
If we multiply by b, we have
b8=zab + Qb* + ab^ + ah^ + aft' +• . • . oft*** and subtract-'
ing the above equation, there remains (6 — i) sasab^ — a;
whence we easily deduce the sum required 8 = — l^; • Conse^
quentlfif the sum ijf any gtamekical pfogrc8$ian U found by miMU
]^ying the la$t term by the exponent eftbe progressionf eubtraeHng
ihejiret term from the productf and dMding the remainder by the
exponent minus unity.
459. Let there b^ a geometrical progression of seven terms,
of which the first = 3 ; and let the exponent be = 2 ; we shall
then have a = S, ft = 2, and it ^ 7 ; wherefore the last term =
3x2*,orSx64= 192; and the whole progression will be
S, 6, 12, 24, 48, 96^ 192.
Further, if we multiply the last term 192 by the exponent 2,
we have 384; subtracting the first term there remains 3^1 ;
and dividing this by ft — 1, or by 1, we have 381 for the sum of
the whole progression.
460. Again, let there be a geometrical progression of fiix
terms ; let 4 be the first, and let the exponent be = |. The
progression is
^, D, 2f, -{ 9 7 9 7 •
If we multiply this last term 'f ^ by the exponent |, we shall
have y/ ; the subtraction of the first term 4 leaves the remain-
der Y/ , which divided by ft — 1 = |, gives *!* = 8S|^.
46U When the exponent is less than 1, and consequently,
when the terms of the progression continually diminish, the sum
of such a decreasing progression, which would go on to infinity,
may be accurately expressed.
For example, let the first term = 1, the exponent = |, and
the sum = s, so that
« = 1 + i + ^ + i + tV + 3V + tV + &c.
ad infinitum.
«
159 tl^ie&nu Sect 3.
If we raidtiply by it, we haye
ad infinituin.
And, subtracting the preceding progressitny there remains
s = £ for the sum of the proposed infinite prog^'esston.
462. If the first term = 1, the exponent = ^, and the sum
s s : so that
« = 1 -f. I 4- { + ,\. + jif -f. &c. ad infinitum.
Multiplying the whole by S, we have
S«=:3 + l+^ + | +^V + ^* ^^ infinitum ;
and subtracting the value of s, there remains 29 = 3; wherefore
the sum ^ = if*
463. Let there be a progression, whose sum = «, first term
= S, and exponent = ^ ; so that s = 2 + |4-|+4|4- ^^ +
&c. ad infinitum.
Multiplying by |, we have4« = 1 + 2+1 + • +|T^^i^
+ &u ad infinitum. . Subtracting now the progression s, there
remains ^ $zz^; .wherefore the sum required = 8.
464. If we suppose, in general, the first term = a, and the
exponent of the progession = — , so that this fraction may be
c
less than 1, and consequently c greater than (; the sum of the
progression carried on^ ad infinitum, will be found thus ;
„ . , at . a6« ofc* . oft* ^
Make.3 = a+ -+ — + ^^+_+&c.
Multiplying by — » we shall have
c
— « = — -i — r- -4 — - -A — 7- + &c. ad infinitum.
And, subtracting this equation from the precedingt there re-
mmns (1 — ^)g^a.
Conseqoentl J < » ^^
c
If we multiply both terms of this fraction by c, we have
3= — The sum of the infinite geometrical progression
Chap. 10. Ratios and Froportum. ^ 151
propoaed is, therefore, found by dividing the first term a by 1
minus the exponent, or by multiplying the first term a by tlie
denominator of the exponent, and dividing the product by the
same denominator diminished by the nume^torof the exponent*
465. In the same manner, we find the sums of progressions,
the terms of which are alternately a^ted by the signs + and
-— • Let for example,
ab c6* fl6* lift*' p
c ' c* c* c*
Multiplying by — , we have
c
e ^ c "^ "c*""*" c' c*
h
Aiy(^i^ adding to this equation to the preceding, we obtain (1 +— ) s
a
ss a. Whence we deduce the sum required s = YIIV ®^ * ==
ae
c + ft'
466. We see^ then, that if the first term a = I, and the expo-
nent = \f that is to say, ft = £ and c= 5, we shall find the sum
of the progression 1 4. /,. -f. ^y^ + /^j -{. &c. = 1 ; since, by
subtracting the exponent from i, there remains I* and by divid-
ing the first term by that remainder, the quotient is 1.
Further, it is evident, if the terms be alternately positive and
negative, and the progression assume this form ;
the sum* win be
^^ ;r~ ""■ "t^«
c
46r. Another example. Let there be proposed the infinite
progression,
The first term is here y'^, and the exponent is y\f. Subtract-
ing this last from 1, there remains ^^^ and, if we divide the
first term by this fraction, we have | for the sum of the given
progression So that taking only one term of the progression,
namely -^^ the error would be -^V*
Taking two teraw ^+^^^^V^Mn wooU fOBL be
wanting tiv ^'^ Aak« the stm = |.
46a» riflnofAer txamfUm Let there be piren tbte iniBite pro*
The lirat term vt % the exponent » •^. So tb«f 1^ niaos tin
exponent^ = iV ^ aiid^ = 10 the sum required.
This series is expressed bja decionl fradioiiy tbns 9,9999999^
CHAPTER XL
Cf Infinite Decimal Fractions.
469. It will be very necessary to shew how a Tulgar fraction
may be transformed into a decimal fraction ; and, conTersely,
how we may express the value of a decimal fraction by a vulgar
fraction.
470. Let it be required, in generalf to diangeihefnuAm -^^ tiilo
a decimal fraction ; as this fructian expresses the quatiemt of the
division of the nvmerator sl In/ the denomvudor h, let us roriie^
instead of a, the quantity a^OOOOOOO, whose value does not at all
difftrfrom that of a« since it contains neitiier tenth parts, nor Aim-
dredthpartSf ^c. Let us funv divide this quantity by the number hf
according to the commonrules of dlMxixm, observing to put the point
in the proper place, ichich separates the dedmal and the integersi
This is the whole operation^ which we shall illustrate by some
examples.
Let there be given first the fraction i, the division in deci-
mals will assume this form,
g) 1,0000000 _1
0,5000000 "" 2*
Hence it appcarSf that | is equal to 0,5000000 or to 0,5 ;
which is sufficiently evident, since this decimal fraction repre-
sents /^9 which is equivalent to f.
Chap. 11. Of BaUt»and Propcfium. 1S3
47U Let ^ be the given fraction^ and we liave,
S) 1,0000000 . _1_
0,3333333 ^* ""S *
This shews, that the decimal fraction, whose value is=^^»
cannot/ strictly^ ever be.discontinaed, and that it goes on ad
infinitum, repeating always the number 3. And, for this reason,
it has been already shewn, that the fractions •^\ + ^|^ + ^^^
+ Tirfirr ^« ^ infinitiimf added together, make ^
The decimal fraction, which expresses the value of |, Is also
continued ad infinitum ; for we have,
3) 2,0000000 , _£
0,6666656 3'
And besides, this is evident from what we have just said,
because | is the double o(^.
472. If i be the fraction proposed, we have
4) 1,0000000 ^ _ 1
0,2500000 ^'""T
So that } is equal to 0,2500000, or to 0,25 ; and this is evi-
dent, since ^^ + 7*7 = tVv = i*
In like manner, we should have for the fraction |,
4) 8,0000000 _ 8
0,7500000 ■" 4 *
So that I = 0,75 5 and in fact 7V + TVT = tW = i-
The fraction 4 is changed into a decimal fraction, by making
4) 5,0000000 _ 5
l,2o00U00 ■" 4*
Nowl+^«^=i.
473. In the same manner, | will be found equal to 0,2; | =
0,4; 4 = 0,6; | = 0,8; 4=1; 4 = 1,2, &c.
When the denominator is 6, we find 4 = 0,1666666, &c. which
is equal to 0,666666 — 0,5. Now 0,666666 = 4, and 0,5 = 4,
wherefore 0,1666666 = f ~ 4 = 4.
We find, also, 4 = 0,833883, &c. = 4 9 ^^^ i becomes
0,5000000 = 4. Further, 4 = 0,833333 = 0,383333 -f 0,5,
that is to say, 4 -f. 4 = 4.
474. When the denominator is 7, the decimal fractions be-
come more complicated. For example. We find 4 = 0,142857,
however it must be observed, that these six figures are repeated
EuLJBg. 20
154 Jtgdfra. Sects.
contiDuall/. To be cotiTinced, thertforef that fhis decioud
fraction precisdy expresses the value of ^f we may transfomi if
into a geometrical progression^ whose first term is = -r VdV aVf
and the exponent = TTvVvinrf ^^f conaequenUy^ the sum
(art. 464)= -^^^J-^ (multiplying both terms by 1000000)
"^ PfVf HI ■*" T'
475. We may prove, in a manner atiU more easy, that the
decimal fraction which we have found is exactly a ^ ; for sub-
stituting for its value the letter s, we have
« = 044d85n4e85714285rf &c.
lOsszl, 428571428571428579 &£• «
100 fs 14, 2857142857142857, ftc.
1000 8 = 142, 857142897142857, &C.
10000 • s 1428, 57142857142857, &C*
100000 S = 14285, 7142857142857, &C.
1000000 8 = 142857, 142857142857, &C.
Subtract «= 0, 142857142857, &c.
999999 8 ^ 142857.
And, dividing by 999999, we have 8 = iftrlv = t* ^b^i^-
fore the decimal fraction, which was made x: s, is = |.
476. In the same manner ^ may be transformed into a deci.
mal fraction, which will be 0|28571428, &c. and this enables us
to find more easily the value of the decimal fraction which we
have supposed =s «; because 0,28571428 &c. must be the double
of it, and, consequently, = 2 s. For we have seen that
100 a = 14,28571428571 &c.
So that subtracting 2 • =: 0,28571428571 &c.
there remains 98 s ss 14
wherefore s = || s |«
'We abofind | = 0,42857142857 &c. which, according to our
supposition, must be = 3 s ; now we have found that
10 s = 1,42857142857 &C,
So that subtracting 3 9=: 0,42857142857 &c,
we have 7 9 = 1, wherefore < s= |.
Chap. 11. Of BaHoB and ^roporiUm. > Iff
477.. When a proposed fractton^ therefore, has tlie dmiMllfta-
tor 7, the dectmid fraction is infinite, and 6 figom are con^
tinually repeated. The reason is, as it is easy to perceive, that
when we continue the division we most return, sooner or later,
to a renainder which we have had ahready. Now, in this diiris*
ion, 6 different numbers only can form the remainder, namely#
1, ft, 3, 4, 5, 6 ; so that, after the sixth diviaion, at furthest, the
same figures must return ; but when the denominator is such as
to lead to a division without remainder, these cases do n A
hai^^eii.
478. Suppose, now, that 8 is the denominator of the fraction
proposed i we shall find the following dedmal flractions ;
\ as 0,185 ) 4 s 0,29; 4 s: 0,37f ; | ±s 0,1 ; f s 0,625 $
f s 0,75 ; 4 s 0,875 ; &c.
If the denominator be 9, we have \ » 0,li^pc« % a 0,882
&C. I = 0,333 &c. «S
If the denominator be 10, we ^ s: 0,1 ; -^^ 0,2 ; ^^ =
0,3. This is evident from the nature of the thing, as also that
^ = 0,01 ; that ^Vir = 0>37 ; that tVA = ^56 ; that ^^ft^
=: 0,0024 &c.
479. If 11 be the denominator of the given fraction, we shall
have ^ = 0,0909090 &c. Now, suppose it were required tb
find the value of this decimal fraction ; let us call it s, we shall
have • s 0,090909, and 10 $ » 00,909090 ; further, loa $ s;
9,09090. If, therefore, we subtract from the last the value of s,
we shall have 99 s = 9, and consequently « = ^V = yV* ^®
shall have, also, ^^ = 0,181818 Ac. j ^^ = 0,272727 Ac. ; ^^ =
0,545454 &c.
480. There is a great number of decimal fractions, therefore,
in which one, two, or more figures constantly recur, and which
continue thus to infinity. Such fractions are curious, and we
shall shew how their values may be easily found.
Let us first suppose^ that a single figure is constantly repeat-
ed, and let us represent it by a, so that « s Ofodaaaaa. We have
10 « = OfOaaaaaa
and subtracting s = Q,aaaaaaa
we have 9 s=sa} wherefore « = -^^
9
ise MgOra. Sect 3.
When two figures are rqieated, as aly we have s = Otdbabaiuu
Therefore 100 « = ab/ibabab; and if we subtracts from it, there
remains 99 s = a 6 5 conseqaently 9:s —,
yfhen three figures^ ba abc, are foun4 repeated* ^ have
s » OyOftcaioaftc ; consequently, 1000 » =s aAc^akoic ; and sub-
tract s from it^ there remains 999«s=afrc; wherefore « =
mbc ,
999 • *"^ "^ ™-
WhenoTer, therefore, a decimal fraction of this kind occurs^
it is easy to find its value. Let there be given, for example^
0,296296, its value will be ||( = ^V' dividing both terms by 27^
This fraction ought to give again the decimal fraction pro-
posed ; and we may easily be convinced that this is the real
result, by dt vigjng 8 by 9, and then that quotient by S, because
Sr n 3 X 9.9|re have
9) 8,0000000
3) 0,8888888
0,2962962, &C.
which is the decimal fraction that was proposed.
481. We shall give a curious example by changing flie frac-
tion r—T-—r—Tr--7*^-s — ^00 ,^ »intoa decimal frac-
1X^X3x4x5x6x7x8x9x10'
tion* The operation is as follows.
2) 1,00000000000000
d) 0,50000000000000
«M
4) 0,16666666666666
5) 0,04166666666666
6) 0,00833333333333
7) 0,00138888888888
t
Chap. 1 1. Of JStatiof mul rtofrnVm. vn
<
8) 0»00019841269841
9) 0,0000£4801 58730
10) OjOoodosrssrsiga
■««
0,00000087557319.
SECTION IV.
OF ALOEBRAIO SqUATIONS, AlTD OF THE KESOLUTION OF THOSE
* SqUATIONI*
CHAPTER I.
Of the Solution of Problems ingeneraL
ABTICUS 482.
7he principal object of Algebra, as well as of all the parts
of Mathematicsy is to determine the value of quantities which
were before unknown* This is obtained by considering atten-
tivdj the conditions given, which are dways expressed in
known numbers. For this reason Algebra has been defined,
1%6 science which teaches how to determine unknown quantities 6y
means of known quantSies,
483. The definition, which we have now given, agrees with all
that has been hitherto laid down. We have always seen the
knowledge of certain quantities lead to that of other quantities,
which before might have been considered as unknown.
Of this, addition will readily furnish an example. To find
the sum of two or more given numbers, we had to seek for an
unknown number which should be equal to those known num*
bers taken together.
In subtraction we sought for a number which should be equal
to the di8*erence of two known numbers.
A multitude of other examples are presented by multiplica-
tion, division, the involution of powers, and the extraction of
roots. The question is always reduced to finding, by means
of known quantities, another quantity till then unknown.
484. In the last section also, different questions were resolved,
in which it was required to determine a number, that could not
Cbap. 1. qfSquationa. 15».
be deduced from the knowledge of other g^ven nmnbeny 'except
under certain conditions.
All those questions were reduced to findings by the aid of
some given numbers, a new uomber which should have a certain
connexion with them ; and this connexion was determined by
certain conditions^ or properties^ which were to agree with the
quantity sought
485. When we haoe a question to reeotoe, we regrestni tHemm-
her eoughi by one of the last letUre ef the alphabet, amd then csn-
rider in what manner the given condUiom can farm an equality
between two quaniUies. This equalityf which is represented by
a kind of formufa, called an equation, enables us at last to deter-
mine the value of the number sought, and consequently to resolve
the question. Sometimes several numbers are sought; but
they are found in the same manner by equations.
486. Let us endeavour to explain tiiis further by an example.
Suppose the following question, or prMem was proposed.
Twenty persons, men and women, dine at a tavern ; the sham
of the reckoning for one man is 8 sous,* that for one woman is
7 sous, and the whole reckoning amounts to 7 livres 5 sous ;;
required, the number of men, and also of women ?
In order to resolve this question, let us suppose that the num*
her of men is = a? ; and now considering this number as known^
we shall proceed in the sam^ manner as if we wished to try
whether it corresponded with the conditions of the question.
Now, the number of men being = x, and the men and women
making together twenty persons, it is easy to determine the
number of the women, having only to subtract that of the men
fi*om 30, that is to say, the number of women = 20 — - x.
But each man spends 8 sous ; wherefore x men spend 8 x sous.
And, since each woman spends 7 sous, 20 — - x women must
spend 140 — 7x sous.
So that adding together 8 x and 140 — 7x, we see that the
whole 20 persons mast spend 140 + x sous. Now, we know
already how much they have spent; namely, 7 livres 5 sous,
or 145 sous ; there must be an equality therefore between 140
* A 8001 18 ^ of a line i a livte | oft cronrii, or 17 centf 6 milla.
160 JKgtira. Sect 4.
4- X and 145 ; that is to say, we have the equation 140 + a? =
1459 and thence we easily deduce x=z5.
So that the company consisted of 5 men and 15 women.
487. Mother qatsHon of the namtkini.
Twenty personsy men and women, go to a tavern ; the men
spend 24 florins, and' the women as much ; but it is Tound that
each roan has spent 1 florin more than each woman. Required,
the number of men and the number of women ?
Let the number of men = x»
That of the women will be = 20 — x.
Now these x men having spent 24 florins, thq share of each
man is — florins.
X
Further, the 20 -7- x women having also spent 24 florins, the
24
share of each woman is florins.
20 — X
But we know tliat the share of each woman is one florin less
than that of each man ; if, therefore, we subtract 1 from the
share of a man, we must obtain that of a woman ; and conse-
24 24 *
quently — — 1 = — . This, therefore, is the equation from
which we are to deduce the value of or. This value is not found
with the same ease as in the preceding question ; but we shall
soon see that a; = 8, which value corresponds to tlie equation ;
for V — 1 = T¥ includes the equality 2 = 2.
488. It is evident how essential it is, in all problems, to con-
sider the circumstances of the qae^tion attentively, in order to
deduce from it an equation, that shall express by letters the
numbers sought or unknown. After that, the whole aii consists
in resolving those equations, or deriving from them the values
of the unknown numbers ; and this shall be the subject of the
present section.
489. We must remark, in the first place, the diversity which
subsists among the questions themselves. In some, we seek
only for one unknown quantiy ; in others, we have to find two,
or more ; and it is to be observed, with regard to this last case,
that in order to determine them all, we must deduce from the
circumstances, or the conditions of the problem, as many equa-
tions as there are unknown quantities.
Chap. 2. OfEquaiums. 161
490. It must have already been perceived^ that an equation
consists of two parts separated by the sign of equality^ => to
shew that those two quantities are equal to one another. We
are often obliged to perform a great number of transformations
on those two parta^ in order to deduce from them the value of
the unknown quantity ; but these transformations must be all
founded on the following principles ; that two quaniiJties remain
equal, whether we add to them, or subtractfrom them equal quanA'
ties ; whether we muliiply them, w divide them by the same num-.
her ; wliether we raise them both to the same power, or extract
their roots of the same degree.
- 491. The eqifations^ which are resolved most easily^ are those
in which the unknown quantity does not exceed the first power,
after the terms of the equation have been properly arranged ;
and we call them simple equations, or equations ofthejirst degree.
But \(, after having reduced and ordered an equation, we find in
it the square, or tlie second power of the nnknowB quantity, it
may be called an equation of the see(md degfUf which is more
di£Bcult to resolve*
CHAPTER 11.
Of the Besdutum of Simple Equations, or Equations of the first
degree.
49£. When the number sought, or the unknown quantity, is
represented by the letter x, and the equation we have obtained
is such, that one side contains only that x, and the other simply
a known number, as for example, x = 25, the value of x is
already found. We must always endeavour, tlierefore, to
arrive at such a form, however complicated the equation may
be when first formed. We shall give, in the course of this
section, the rules which serve to facilitate these reductions.
493. Let us begin with the simplest cases, and suppose, first,
that we have arrived at the equation a? + 9 = 16j we see imme-
diately that xzz7. And, in general, if we have found x + a
= b, where a and ( express any known numbers^ we have only
Evl. Jilg. 21
16£ Jtg^bNU 8Mt4.
to BubtniQt a from both sides^ to obtain flic oqvatioD « = i — p^ a^ '
which iodicateo the yalae of or,
494. If the equatipti which we have {bmid isx — a s i^ we
ai)4 a to both «ide0» and obtain the value q^ x^h + a.
We proceed in the aame manner, if the equation has this
form^ X'-^a^aa-k'li for we ahaH have immediateljr » s aa
+ tt+l.
In fliis equation* X — aaavSO— -6a, we find xr^jljO-— 6f
-f. 8 a, or a?s: SO -f ft «•
And in this* JC + 6a=ftO-f da, we havea^sftO-fSo— 6a^
or 0? =: £0 -— 3 a*
495. If the original equation has this fonOf a? — -f 6 s c^
we may begin by adding a to both aidea, which will give x -f
h9s€ + a; and then subtractiiig b frpm both sides, we abaH
Snd x^e+a — i. But we might also add +a — h at once to
both sides ; by this we obtain immediately x^c + a^^h.
So in the following examples,
Ifo:*-— ftn-f 36 = 0, we have xsfta— -3fc.
If 07 — 3a-f-ft& = S5 + a + 2ft, we hav^ :rs=25-|-4««
If a:* — 9 -f 6a = 25 +£a, we havear=K34 — 4a.
496. When the equation which we have found has the form
a OP = &, we only divide the two sides by a, and we have a? = -•
Bnt if tlie equation has the form aa;+&— .o=:d,we must first
make the terms that accompany a x vanish, by adding to both
sides ^— ft -I- c ; and then dividing the new equation, a xp = d — -
& -f c, by a, we shall have x = -HI— Xf.
n
We should have found the same value by snbtracting Tf t— ^
from the given equation ; that ia^ we should have had, in the
same form, aa? = d^-6-fc, and x = "^^ Hence,
V%x + 5^l7f we have 2:vssldf anda9s=6.
If 3 07— 8 = 7, we have 3 a? ss 16^ and aeiss 5.
If 4«-^5 — 3a:9l$ + 9(if wehave4a;sft0-f l^OfandfCon*
sequently, a? s= 5 + 3 a.
49r. When the first equation has the form - = ft; we multiply
both sides by a, in order to have ;c =: a ft.
Chap. d. qfSquaSant. 163
But if we have — 4. ft— c = d, we must first make — = i2 -^ ft
a a
-f-c; after which, wefind a=:((I — b + c)a^ad — ab + c
t^t ^ X — 9 =: 4, we have | a? =2 /, and aJ =s 14.
Let |ar-^l-fda=:d-f^a|We have | « s: 4 -^a^ and a? s
id -^ d a.
I^t r — 1 = a, we have ' =:a + 1, and or = a a -- 1.
A **^ I a ■■• 1'
»
adr
498. When we have arrived at such an equation Sis -r- =c,
9
ita f rst ault^ly by ft, in order to have ax^hCf and then divid-
ing hy Qf we find or = — •
If fL5 — . c = d, we begin by giving the equation this form *-i?
ft ' 'ft
s=d-f-C9 after which we obtain the value of aa7 = ftd + ft e, and
fliat of :ra= — -i— £.
I^et U8 suppose \x — 4 =s 1» we shall have | ;r = 5^ and d x
= 15 ; wherefore x^sz y , or 7|.
If I X +\ = 5, we have | x = 5 — | = |^ wherefore 3 x =
18, and x = 6.
409. Let ds now consider the case, which may frequently
occur, in which two or more terms contain the letter Xf either
on one side of the equation or on both.
If those terms are all on the same side, as in the equation x -f
^ 0? -f 5 = 1 1, we have a; -f | a; = 6,or 3ar=: 12, and, lastly, x =s4.
' Let xc + ^x-f-^a7=44, and let the value of x be required :
if wefiri^ multiply l>y 5, we h«ve4a^4.|a?=?l:32$ tb^n nralti-
pl> ing by S, we have 110:=: 264 ; wherefore or = 24. We might
have proceeded more shortly, beginning with tile reduction of
the three terms which contain a;, to the single term yo?; and
th^n dividing tlie equation yx=44 by 11^ we^ooM have
had I or = 4, wherefore a: = 24. ^
Letfx— |;v^|«=BVwe riudl have^ byrtod«Ofion^ ^x
= 1^ and a; = 2}.
164 «%dra» Sect 4«
Letf more generally^ a or — bx + cxsid; this is the same as
(a — & 4- c) 0? = di whence we derive x = r— — .
^ "^ a-^b+ c
500. When there are terms cwtaining x on both .sides of the
equatipUf we begin by making such terms vanish from the side
from which it is most easily done ; that is to say^ in which there
are fewest of them.
If we have, for example, the equation Sx + Z^^x + lO, we
must first subtract x from both sides, which gives ^ a; -f 2 = 10 ^
wherefore S a; = 8, and a? = 4.
Leta; + 4 = 20 — x; it isevident that £a? + 4 = £0; and
consequently £ a: = 16, and a? = 8.
Let 0? + 8 =: 32 — 3 a;, we shall have 4a; + 8 = 32 : then Ax
= 24, and OF s= 6.
Let 15 — a: = 20 — 2x, we shall have 15 +;i? = 20, and
x=i5.
' Letl+a;s=5 — |a?, w^shallhave l+|a;s:5^ after that
|a; = 4; 3a? = 8; lastly, a: = 1 = 2 1.
If I — 3^ = i — i^p we must add I a:, which gives |sj-f
^j x$ subtracting |, there remains ^^ x = j^'aifd>nultiplying by
12, we obtaiQ a:= 2. .!.. * .
If H — 4a: = J + ia:,weadd|a:,whichgif^li=i + Ja:,
Subtracting ^, we have |- a: = l|, whence we deduce x = 1-^ s
^|, by multiplying by 6, and dividing by 7.
501. If we have an equation, in which the unknown number
0? is a denominator, we must make the fraction vanish, by multi-
plying the whole equation by that denominator.
Suppose that we have found — — 8 =: 12, wo first add 8^ and
X
have — r cSO j then multiplying by or, we have 10t>a:%bi?;
X
and dividing by 20, we find a? s 5.
jt we multiply by a: — 1, we have 5a: + 3=:7a?— 7.
Subtracting 5 or, there remains 3 = 2 ar — 7. '
Adding 7f we have 2 or = 10. Wherefore x^5.
' i
Chap. S. OfBquahons. 109
50S. SometimeB^ also^ radicid signs are foond in eqnatbnB of
the first degree. For example^ a number x below 100 is re-
qaired, and snch* that the square root oC 100 — x may be equal
to 8, or V(ioo— X) = 8 ; the oqoare of both sides will be 100
—. :r = 64y and adding x we ha?e 100 s: 64+ a:; whence we
obtain « 3= 100 — 64 = 36.
Or^ since 100 — or = 64^ we might have subtracted 100 from
both sides ^ and we should then have had — rr = — < 36 ; whence
multiplying by — 1^ 0?= 36. .
CHAPTER in.
Of the Solution of Questions rdating to the pr^peding chapter*
50S. questimi I. To divide 7 into two such parts, that the
greater may exceed the less by S.
Let the greater part = x, the less will be = 7 — x; so that
Xzs7 — x + S, or ar=slO-— x; adding x, we have 2 a? = 10;
and, dividing by 2, the result is a? = 5.
Jnswer. The greater part is therefore 5, and the less is 2.
Question II. JHf^ required to divide a into two parts, so that
the greater may exceed the less by ft. x
Let the greater part = x, the other will be a -— or ; so that
x^a — a:-f6; adding or, we have £ar = a +ft; and dividing
ty«^a?=-i-.
^ Aiother Solutiaiu Let the greater part = or ; and, as it ex-
ceeds the less by b, it is evident that the less is smaller than the
other by h, and therefore must be = a; — » ft. Now these two
parts, taken together, ought to make a; so that2a?'^6=a;
adding ft, we have 2 a? = a + 6, wherefore x = ^^-^^ which is
the value of the greater part ; that of the less will bfllMlz .^ ft^
a + h 2ft a— .ft
or— J -., or—- — •
2 3 2
504. iljoestian III. A father, who has three sons, leaves them*'*
1600 crowns. The will specifies, that the eldesrt shall have soo
166 J h rt i iu Sect. 4.
crowBB more fhan the accom^ aad fhat the Mcmid sliftll bave
100 rrowm More tban the yoongeet Beqoifed the share of each ?
Let the share of the thM eon ^x; then, that of the eeeontf
waibesor+lOO^and that of the ftret 3=9 + 300. Now these
fliree shares make 19 together 1000 crowus. We hav^
therefore.
Sjr + 400s±l600
and X =z 400.
Jkinver. The share of the youn^i^est is 400 crowns ; that of
the second is 500 crowns ; and that of the eldest is 700 crowns.
505. (fueslumlV* A lather haves four sons, and 8600 livrcs;
according to the wiU, the share of the eldest is to be douhle that
of the second, minns 100 litres ; the second is to recejye three
times as much as the third, minos 800 Kyrrs ; and the« third is
to receive foor times as much as the fourth, minus 500 livresb
Bequired, the respective portrana of these four sons.
Let us call x the portion of the yomigetit ; that of the third
sonwiU be = 4ar— -SOO; that of the seeoad = l£a?—- 1100,aad
that of the eldest s? 34 at^^ 2doa The earn of these four shares
mast make 8600 Httbs. We have, therdbre,^^e efoatiou 41 x
— 3700 = 8600, or 4 1 a; = 1S300, and x = 300«
Jtnnotr. The youngest must have 300 livres, tim ttiird i(m
900, the second 2500, and the eldest 4900^
506* ^estum. Y. A man leaves 1 1000 crowns to he divided
between his widow, two sons, and three daughters. He intends
that the mother should receive twice the share of a;8oa,aiideach
son to rccrive twice as naich as a' dougjhter*' Binaiwjij • heiw
much each of them is to receive I
Suppose the share of adauff^erssj^ thaiof asoais couaa*'
quently = ex,aad that of the nvidow = 4 x; the whole inheritaiice
is therefore t^x + Ax + ^x ; so that \lx=^ 11000, and x =: 1000.
AnsweufV.9c\i daughter receives 1 000 crowns^
So that the three receive in all 3000
Each son receives fiOOO crowns.
So that both the sons receive 4000
And the mother receives 4000
Sam 11000 crowns.
Chap. S. QfiSfiMitfofU. 167
507. questum VI. A father intends, by Us wfll, (hat bis
three sons should share bis property in the following manner ;
the eldest is to receive 1000 crowns less than half the whole
fortune; the second is to receive 800 crowns less than the third
of the whole property j and the third is to have 600 crowns less
than the fourth of the property. Requiredf the sam of the whole
fortune^ and the portion of each son 2
Let us express thelhrtune by x.
The share of the Brst son is ^ x — 1000
That of the second 4^— BOO
That of the third ^x^ 600,
So that the three sons receive in all Ix+^x+lx-^Q^OOf
and this sum must be equal to cc.
We have, therefore, the equation 41 ^ *^ ^^ = ^«
Subtr^ing x, there remainsi ^^ x — 2400 = 0.
Adding 2400, we have ^j x = 2400. Lastly multiplying by
12, the product is x equal to 28800. '
Jlnswer. The fortune consists of 28800 crowns, and
The eldest of the sons receives 13400 crowns
The second 8800
The youngest 6600
28S00 crowns.
508. HuegtUm yiL A father leaves four sons, who share his
property in the following manner :
The first takes the half of the fortune, minus SOOO livres.
The second takes the third, minus 1000 livres.
. The third takes exactly the fourth of the property.
The fourth takes 600 livres, and the fifth part of the property.
"What was the whole fortune, and how much did each son
receive ?
Let the whole fortune be = d? ;
The eldest of the sons will have ^ a? — SOOO
The second 1 op — 1000
Thetliird Ix ^
The youngest t ^ + 600.
The four will have received in all ^x + }x + ^x + ix —
S400> which must be equal to x.
168 JUgOra. Sect 4.
Whence results the eqttation Hx^^ S400 = x ;
Subtracting x, we have ^x — 3400 = ;
Adding 3400, we have Hx^ 3400 ;
Dividing by 17, we have ^V ^ = ^^^ '*
Multiplying by 60, we have ;r = IdOOO.
Jhuwtr. The fortune consisted of 12000 livrcs.
The first son received SOOO
The second 3Q00
The third 3000
The fourth 3000
509« (fuestion YIIL To find a number such, that if we add
to it its half, the sum exceeds 60 by as much as the number itself
is less than 65,
Let the number = 079 then 0? -f 1 0? — 60 = 65 — 07 ; that is to
say ^x — 60 = 65 — « a; ;
Adding x* we have ^x — 60 = 65 ;
Adding 60, we have | cc = 125 ;
Dividing by 5, we have ^ x = 25 ;
Multiplying by 2, we have x = 50.
Jlnswer. The number sought is 50.
510. ^ustion IX. To divide 32 into two such parts, that if
the less be divided by 6, and the greater by 5, the two quotients
taken together may make 6*
Let the less of the two parts sought =x; the greater will be
= 32 — x; the first, divided by 6, gives ^ ^ the second, divid-
ed by 5, gives ~ ; now, - + . T" =6. So that multiply*
O O D
ing by 5, we have ix + S2 — a? = 30, or — | cr -f 32 == SO.
Adding | x, we have 32 = 30 -f- j- ^.
Subtracting 30, there remains 2 = | x.
Multiplying by 6, we have a? = 12.
Mswer. The two parts are ; the less =: 12, the greater = 20.
511, (futstUm^. To find such a number that if multiplied
by 5, the product shall be as much less than 40, as the number
itself is less than 12.
Let us call this number x. It is less than 12 by 12 — x.
Taking the number x five times, we have 5 x^ which is less than
40 b|f^ ^-«» 5 X, and this quantity must be equal to 12 -» 0^4
«
i
Chap. 5. O/S^pusHmu 169
We have therefore 40*— 5 « = 13 — ^r.
Adding 5 x, we have 40 =s IS +Ax.
Subtracting 12, we have 28 = 4 a?.
Dividing by 4» we have « =: 7f the number sought.
512. ifuestion XL To divide 25 into two such parta/that the
greater may contain the less 49 times.
Let the less part be s= iTy then the greater will be = 25 — x.
The latter divided by the former ought to give the quotient 49 ;
25 —X
we have therefore = 49.
X
• Multiplying by Xf we have 25 — a? = 49 x.
Adding x 25 == 50 x.
And dividing by 50 a? =:|.
Answer, The less of the two numbers sought is ^9 and the
greater is 24^ ; dividing therefore the latter by I, or multiplying
by %9 we obtain 49.
513. Huestion XIL To divide 48 into nine parts, so that
every part may be always ^ greater than the part which pre-
cedes it.
Let the first and least parts; a?, the second will be = a? + ^,
the third = a? + I» &c.
Now these parts form an arithmetical progression, whose
first term =2;; therefore the ninth and last will be=:a?-f-4*
Adding those two terms together, we have 2 a; +4; multiplying
this quantity by the number of terms, or by 9, we have lSx +
36 ; and dividing this product by 2, we obtain the sum of all
the nine parts = 9 a; + IB ^ which ought to be equal to 48. We
have, therefore, 9 a? + 18 = 48.
Subtracting 18, there remains 9 ^ = 30. ^
And dividing by 9, we have a? = 3^.
Jinswer. The first part is 3^, and the nine parts succeed in
the following order :
1234567 89
Si + S|+4^+4J + 5^+5| + 6| + 6f + 7i
w*hich together make 48.
514. ^uestioii XIII. To find an arithmetical progression^
whose first term = 5, last =10, and sum = 60.
Here, we know neither the difference^ nor the nonlier of
Bui. Jttg. 22
JUgebrom 80ct»4»
terms ; but we know that the first and the last term -WQuld
ble us to express the sum of the progressioat provided 9oXy the
number of terms was given. We shaU* tbereforef siippoae this
tKjg
number = x^ and express the sum of the progression hj -^ •
now we know also that this sum is 60 ; so that ..--« » 60 s Ix
X . *
= 4, and x = 8«
Nowy since the number of terms is 8^ if we suppose the diflfer-
ence = »» we have only to seek for the eighth term upon this
supposition, and to make it = 10. The second term Ib 5^%,
the third is 5 + S $&« and the eighth is 5 + r s^ ; so that
5 + 7% = 10
7»= 5
and 2C = |.
Answer. The difference of the progression is, ft and the
number of terms is 8 j consequently the progressbn is
123 45 67 8
5 + 5|+6^ + 7| + 7f + 84 + 9^+10
the sum of which = 60.
515. Quesfto?! XIY. To find such a number^ that if 1 be
subtracted from its double, and the remainder be doubled, then
if 2 be subtracted, and the remainder divided by 4, the number
resulting from these operations shall be 1 less than the number
sought. _
Suppose this number = a? ; the double is 2 x ; subtracting I,
there remains S a: — 1 ; doubling this, we have 4 a? — 2; mfik-
tracting 2, there remains 4 a? — 4; dividing by 4, we have
X — 1 ; and this must be one less than x ; so that,
^ a; — 1=0? — 1.
But this is what is called an idenKcal equation ; and shews that
0? is indeterminate ; or that any number whatever may be sub-
stituted for it*
516. ^estion XY. I bought some ells of cloth at the rate of
7 crowns for 5 ells, which I sold again at the rate of 11 crowns
for 7 ells, and I gained 100 crowns by the traffic* How much
doth was there ?
Suppose that there were x ells of it ; we must first see how
UDDOS
muchiUh'iloth cost. This is found by tho following proportion ;
Chap. S« OfBquatums* 1)1
If five dk cost 7 crowns j what do x ells cost I
Jhuwer, I x crowns.
This was my expenditure. Let us now see my receipt : we
must make the following proportion ; as 7 ells are to U crowns^
so are a: ells to y x crowns.
This receipt ought to exceed the expenditure by 100 crowns $
we havey thereforef this equation.
yar = |a? + 100;
Subtracting ^ a» there remains ^V ar= 100.
Wherefore 6 a? = S500, and x = 58S|.
JInswer. There were 583^ ells, which were bought for 816|
crowns, and sold again for 916| crowns, by which means the
profit was 100 crowns.
517. ^^e8tion XYI. A person buys 12 pieces of cloth for 140
crowns. Two are white, three are black, and seven are blue.
A piece of the black cloth costs two crowns more than a piece of
the white, and a piece of blue cloth costs three crowns more than
a piece of black. Required the price of each kind 7
Let a white piece cost x crowns ; then the two pieces of this
kind will cost 2 or. Further, a black piece costing x + 2, the
three pieces of this colour will cost Sx + 6. Lastly, a blue
piece costs AT 4- 5 ; wherefore the seven blue pieces cost 7x +
35. So that the twelve pieces amount in all to 12 or + 41.
Mow, the actual and known price of these twelve pieces is
140 crowns ; we have, therefore, 12 n? -f 41 = 140, and 12 a? =
99 ; wherefore or = 8| $
" So that a piece of white cloth costs 8^ crowns ; a piece of black
cloth costs 10^ crowns, and a piece of blue cloth costs 13^ crowns.
518. Question XYIL A man, having bought some nutmegs,
says that three nuts cost as much more than one sous as four cost
biro more than ten Hards : Required, the price of those nuts 7
We shall call x the excess of the price of three nuts above one
sous, or four liards, and shall say ; If three nuts cost x +4
liards, four will cost, by the condition of the question, or -f- 10
tiards. Now, the price of three nuts gives that of four nuts in
another way also, namely, by the role of three. We make 3 : x
4.4 =s4: dnswerp — •^ — •
in JKgeknu 8ecL4^
8othat^-^^^^^=ap + 10; or,4a; + 16sSx+SO; where-
tort X + 16 s=i SO.
and a? ss 14.
Answer. Three nuts cost 18 liards^ and four cost 6 sons ;
wherefore each cost 6 liards.
519. Question XVIII. A certain person has two silver cnps^
and only one cover for both. The first cap weighs 18 ounces^
and if the cover be pat on it, it wei^ twice as macb as the
other cup ; but if the other cop be covered, it weighs three times
as much as the first : Required^ the weight of the second cap
and that of the cover ?
Suppose the weight of the cover = x ounces ; the first cup
being covered will weigh x -f- 12 ounces. Now thi^ weight
being double that of the second cup, this cup must weigh {x+6^
If it be covered, it will weigh | x -f 6 ; and this weight ouglit to
be the triple of 12, that is, three times the weight of the first
cup. We shall therefore have the equation | x-f 6 = 36, or i a;
= 30 ; wherefore | or = 10 and x = ftO.
Answer. The cover weighs 20 ounces^ and the second cop
weighs 16 ounces.
520. €(ue$ti4m XIX. A banker has two kinds of change ;
there must be a pieces of the first to make a crown ;' and there
must be h pieces of the second to make the same sum. A per-
son wishes to have c pieces for a crown ; how many pieces of
each kind must the banker give him ?
Suppose the banker gives x pieces of the first kind ; it is evi*
dent that lie will give c — - or pieces of the other kind. Now, the x
pieces of the first are worth— crown^ by the proportion a : 1 =
a? : — ; and the c — or pieces of the second kind are woHh , ■
a o
crown, because we have h : 1 isic^-^xi "7 * ^ ^^^ -^ +
A
c •— • X hx
— - — s=l; or---- + c— -a;=:(:orft£i?-fac--*aa? = aft; o&
O 8
rather|6a?— -aarsaA— <ac| whence we have x = -r— — » op
aih'^t) ri Ai 6e— lift ft(e«— a)
« = -\- — i. Consequently, c— a = — r = -r ••
o«— a 0— .a — fl
•fln^wer. The banker wiB give ^} '^^V ^^Qf^^^S'^t^^^P^y
and \ "^'^^ pieces of the second kind#
6—11
Bamrk. These two nuipbere are easUy faitfi4:l^y the rule of
three, when it is required to apply the ^iiiults wbidi we haire
ob tained. To find the first we say :6— a:ft — c=a: ; "*^^ ,
The second number is found thus : 6— fl:c— a = 6: \ ^ •
' 6 — a
I^ught to he observed also that a is less than 6, and that e
is also less than b ; but at the same time greater than a, as the
nature of the thing requires*
521. Question XX. A banker has two kinds of change ; 10
pieces of one make a crown, and £0 pieces of the other make a
crown. Now, a person wishes to change a crown into 17
pieces of money : How many of each must he haye ?
We have here a s 10, b = 20, and c = 17 j which furnishes
the following proportions ;
I. 10 : 3 = 10 : d>'so that the number of pieces of the first
kind is 3.
II. 10 : 7 = 20 : 14, and there are 14 pieces of th» second
kind.
522. Question XXI. A father leaves at his death several
children, who share his property in the following manner ;
The first receives a hundred crowns and the tenth part of the
remainder.
The second receives two hundred crowns and the tenth part of
what remains.
The third takes three hundred crowns and the tenth part of
what remains.
The fourth takes four hundred crowns and the tenth part of
what then remains, and so on.
Now it is found at the end, that the property has been divid-
ed equally among all the children. Required, how much it was,
bow many children there were^ and how much each received ?
1T4
JBgdnUm
Sorti* 4«
ThisqiiestkHi'is-rfttlier of a singnlar nature, and therefore
deserves particular attention. In order to resolve it more easiljr,
we shall suppose the whole fortune s x crowns ; and since all
the children receive the same sum^Iet the share of each =: Xf by
which means the number of children is expressed by — • This
bein^ laid down, we may proceed to the solution of the question^
which will be ad foliuws.
SwBf or nro
pert^to M
diridcd.
% — Sap
55 — 4a?
«— 5 X
the
Chililrea
3d.
4th.
5*-
6^
Portkm of omIu
V
0^=100 +
X = 200 +
a: = 300 +
X = 400 +
or = 500 +
a; = 600-f
100
10
10
X — &r — 300
to
a?— aa>--400
X — 4jr— 500
To
x^^^a^ — 600
10
—
— ^. —
^
100-
ON- 100 ^
^ 10 -^
100-
a?— 100
- 10 -^
100-
X— 100 ^
• 10 =^
too-
«— .100 ^
- — — -a=0
10
and so on.
We have inserted, in the last eolumn^ the differences which
we obtain by subtracting each portion from that which follows.
Now all the portions being equalt each of the differences must
be = 0. And a^ it happens that all these differences are express-
ed exactly alike, it >%ill be sufficient to make one of them equal
jp 100
to nothings and we shall have the equation 100 — -
10
= 0.
Multiplying by 10, we have 1000 — x — 100 = 0, or 900 — 00.
s£ ; consequently x = 900.
We now know, therefore, that the share of each ch'ild was
900 crowns ; so that taking any one of the equations of the
third column, the first for example*- it becomes, by substituting
% ■ 100
the value of or, 900 = 100 -) -rz — , whence we immediatdj
obtain the value of %, ; for we have 9000 = 1000 + «— 100, <ir
9000 s 900 -f a; ; wherefore «^ s BlOO ; and consequendy-- s 9*
Answer* So that . flie number of children =9 ; tbe fortune
left by the father =s 8100 crowns ; and tiM share of each child
S.900 crowns.
CHAPTER IV.
Of the Eesdutions (f Iwo (n' nun^fMqiuUim^ of theFkei Degree.
583. It frequently happens that we are obliged to introdoce
into algebraic calculations two or more unknown quantitieSf
represented by tbe letters a?, yf»; and if the question is deter-*
mini^9 we are brought to the same number of equations ; from
which, it is then required to deduce the unknown quantities.
As we consider, at present, those equations only which contain
no powers of an unknown quantity higher than the first, and no
products of two, or more unknown quantities, it is evident that
these equations will all have the form a% + by + ex=zd.
524. Beginning, therefore, with two equationsy we shall en*
deavour to find from them (he values of x and y. That ifd may
consider this case in a general manner, let the two equations be,
Lax + by=:c, and Ihfx+gy =: h, in which a, 6, c, and/, g,
h are known numbers. It is required, therefore, to obtain, fh>m
these two equations, the two unknown quantities x and y,
525. The most natural method of proceeding will readily
present itself to the mind ; which is to determine, from both
equations, the value of one of the unknown quantities, x for
example, and to consider the equality of those two values ; for
then we shall have an equation, in which ihe unknown quantity
y will be found by itself, and may be determined by the roles
which we have already given. Knowing y, we have only to
substitute its value in one of the quantities that express x.
526. According to this rule, we obtain from the first equa-
tion, X = ^~ ^ , and fjrom the second, x = ■ ^^'^ ; statins
^ f
these two values equal to one another, we have this new equa-
tion;
' *
^—by _ h^gy ^
~^ 7"'
176 MgAra. JBecL 4.
midtiplyiiig, by Ot the product is c -— ft y == ~^gy . multiply-
ing by/y the product is/c — ^y &y = a A — -a|r y ; adding a; y» w«
have /c — fby + ugy=iah; subtracting /c, there remains
^fby + agy^ah—fcf or(ag— 6/)y=aA— /cj lastly,
dividing by ag-^bf, we have y = — ^y>
In order now to substitute this value of y in one of the two
ralues which we have found of Xf as iii the first, where x s
^ , we shall first have — 6 y = — . ^"^ i whence c — 6 y
« ^ «g— ft/ ^
acg — abh J ,. ... , e — fty eg — bh
— s — -— ; and dividing by a, a; = -— ^ = -^ — j->
ag — of ° a ag-^bf
5Q7. Question I. To illustrate this method by examples let
it be proposed to find two numbers, whose sum may be = 1$,
and difference = 7.
Let us call the greater number x, and the less y. We shall
have,
I. a; 4- y = 15, and IL x — y = f.
The first equation gives x=zl5 — y, and the second x=z7
+ y; whence results the new equation 15 — y=7'+y. So
thatl5 = 7 + 2y; 2y=8, andy = 4; by which means we find
Answer. The less number is 4, and the greater is 11.
528. ifuesHon II, We may also generalize the preceding
question, by requiring two numbers, whose sum may be = a,
and the difference = ft.
Let the greater of the two be = or, and the less = y.
We shall have I. a? + y =5 a, and IL or — y = ft ; the first
equation gives x=za — y ; and the second x^b + y.
Wherefore a— y = ft + y; a = ft + 2y; 2y = a*-^ft; lastly,
o— ft . ., a— ft £1 + ft
y= —3 — , and consequently X = a — y=:a 5— =— i— .
Answer. The greater number, or x, is =s ^^^ , and the leas,
a — ft
or y, is = , or which comes to the same, xs^la+^b, and
Chsp. 4. Of 3imtim$. 177
j^as|a*-*|(; and hence we derive Ike followiiig theorem.
When the sum of any two mmber^ is a» and thrir HfftUfM u b^
iht griaUr of (As Pwo numhtn vrill be equal to haJ^ the sum plus
kaif the afference ; and the less of the txvo «iiiiiierf ToiH he eqwd
to haff the «uni miiiUB half the iifferenee,
529. We may also resoWe the same question in the followifig
■Mumer;
Since the two equations «rea?-f ysvoy andx^^ysah; if
we add one to the othert we ha?e S x =a+ b.
Wherefore x = "T .
2
Ziaatljry subtracting the same equation from the otber> we have
8y=ia — ft; wherefore y = —^^.
SSO. ifuestion IlL A mole and an ass were carrying iMirdens
amounting to some huadred weight. The aas complained of bi%
and said to the mule^ I need only one hundred weight of yo|iv
loadt to make mine twice as heavy as yours. The mule answefv
ed, Tes9 but if you gave me a hundred weight of yoyrsy I dumld
be loaded three times as much as yoa would be. How many
hundred weight did each carry ?
Suppose the mule's load to be a hundred weight* and that of
the ass to be y hundred weight. If the mule gives one hundred
weight to the ass* the one will have y + If and there will remain
for the other :v— 1 ; and sincOf in this case* the ass is loaded
twice as much as the mulet we have y + l^UX'^Z.
Further! if the ass gives a hundred weight to the mule^ the
latter has x + l, and the ass retains jfi— 1 ; but the burden of
flie former being now three times that of the latter^ we have
OP + lsSy— 3.
Our two equations will consequently be,
L y+lsz^x-^^p II. x+ I s9y-^3.
The first gives or = ^^-—w, and the second gives a?;7 3 y — -4 ;
whence we have the new equation ^"\ = 3 y — 4, which gives
9= Vt and also determines the value of x, which becomes S|.
Jtnswer. The mule carried 2| hundrtiS weight, and the ass
carried Q^ hundred weight.
BuUJUg. 23
178 JUgAra. Sect 4b
5St. When there are three unknown numbers, and as roanj
equations ; as, for example, I.x + jf— «=8, II.a;+« — Jf=9,
III. y + Xf — dP = 10, we begin, as before, by deducing a value
of X from each, and we have, from the I"^, a; a 8 +« — y;
from the 11"^, a;=9+y — »; and from the lU^9X=ijj+c6
— 10.
Comparing the first of these values with the second, and after
that with the third also, we have the following equations ;
I, 8+« — y = 9 + y — », II. 8+!» — y=sy + % — 10.
Now, the first gives £« — 2y=:l, and the second gives 2 y =
18, or y =9 ; if therefore we substitute this value of y in 2%— ^
2 y = 1, we have Zc6 — 18 = 1, and 2 « = 19, so that « = 9| ;
it remains therefore only to determine x, which is easily found
Ql
— .»-y.
Here it happens, that the letter « vanishes in the last equation,
and that the value of y is found immediately. If this bad not
been the case, we should have had two equations between % and
y, to be I'esolved by the preceding rule.**
532. Suppose we had found the three following equations.
]. Sa;-f-5y — 4« = 25, II. 5x — 2y-f d;& = 46,
III. Sy + 5% — x = 62.
If we deduce from each the value of x^ we shall have
25— 5y-f 4;g 46 +2y--3»
I. ^ = — r , 11. X ^ ^ ■
III. x=zSy + 5% — 62.
Comparing these three values together, and first the third
with the first, we have 3y + 5 « -^^2 = £izL£i!L±lf . mrf.
tiplying by3, 9y + 15«— 186 = 25 — 5y + 4«; so that
9y + 15a=:211 — 5y+4», and 14y-f.ll » = 211 by the first
and the third. Comparing also the third with the second, we
45 4-2 t/«^ 3 z
have 3 y + 5 « — 62 = — -^—^ , or46-f2y — d5e=:l5y
+ Q5» — 310, which when reduced is 356 = 13 y + 28 jc.
We shall now deduce, from these two new ec^uations, the
value of y ;
I. 211 = I4y+ ll«j wherefore 14y = 2U — 11«, and
211 — 11 z
y^ — IT—
Chap. 4. Of BquaHons; 179
II. S56=15y + 28»; wherefore 13y = S56~28«;y and
356 — ^Sx
*"" 13
These two values form the new equation
gl1 — lla ;_ 856 — gSa;
14 "" 13 ^
which becomes, 274S — 143» = 4984 — 392 x, or fi49 « = 2241,
whence « = 9.
This value being substituted in one of the two equations of y
and %9 we find y = 8 ; and lastly a similar subcititution, in one of
the three values of x^ will js^i ve or = 7,
5S3. If there were more than three unknown quantities to be
determined) and as many equations to be resolved, we should
proceed in the same manner ; but the calculations would often
prove very tedious.
It is proper, therefore, to remark, that, in each particular
case, means may always he discovered of greatly facilitating its
resolution. These means consist in introducing into the calcu-
lation, beside the principal unknown quantities, a new unknown
quantity arbitrarily assumed, such as, for example, the sum of
all the rest ; and when a person is a little practised in such cal-
culations he easily perceives what is most proper to do. The
following examples may serve to facilitate the application of
these artifices.
534. Question lY. Three persons play together ; in the first
game, the first loses to each of the other two, as much money
as each of them has. In the next, the second person loses to
each of the other two, as much money as they have already.
Lastly, in the third game, the first and the second persoii^. g&in
each, from the third, as much money as they had before. They
then leave off, and find that they have all an equal sum, namely,
24 louis each. Required, with how much money each sat down
to play ?
Suppose that the stake of the first person was x louis, that of
the second y, and that of the third «• Further, let us make the
sum of all the stakes, or x+y + % = 8. Now, the first person
losing in the first game as much money as the other two have,
ke loses e — x; (for be himself having bad x, the two others
180 MgAm. lUfL 4.
iMet have had s^^x); wherefore there wHl renain to Um 9x
— $; the second will have 2 y, and the third will have 2 «»
So tbat^ after the first game, each will have as follows ;
theL2a;~^ thelLSf^ th^IlLfA;*
In the second gamef the second persoiiy who has now S y, loses
as much money as the other two have, that is to say s — 2 y ; so
that be has left 4 y -^ t . With regard to the others^ they iHIl
each have double what they had ; so that after the eecowi ga«B^
the three peremis have |
theL4ar — 3«, the IL 4y— >a» the IIL 4flS.
In the third game, the third person^ who has now 4 «^ is tha
loser ; he loses to the first 4 op -^ 2 s, and to the second 4 y— j;
oonsequenliy after this gaaie the tfiree peraons will have ;
tbeL 8:p---4f, theil«8y^2«t thellLast— ••
Nowy each having at the end of this game 24 louiSf we hava
three equations, the first of which imncdiateiy gives sc, the
second y, and the third » ; farther, t is known to be » 72, ainoe
the three persons have in all 72 ioais at the end of the last gum }
but it is not necessary to attend to this at tnU We have
I. aa:-^4s = 249 or 8x3s244.4«9 orxssS^.^^ ;
II. 8y — 2s = 24, or8y=324 4.2«f orystS-f^t;
III.8«*-^ Ss24, or 8» = 24-|- f, or»3sS<f|«;
Adding these three values, we have
x + y + %^9 + ls.
So tbatf since x+y+»sz$f wo havens 9 + {s; whenfore
I s = 9, and 8 ss 72.
If we BOW substitute this valaa of s in the et pr ess i on s which
we have found for Xf y, and «, we shall find that before Ihegr
began to play, the first person had 39 Ioais ; the second fill Ionia |
and the third 12 louis.
This solution shews, that by means of an expression for tb»
sum of the three unknown qoantitiest we may overcome the
difficulties which occur in the ordinary method.
555. Although the preceding qoestion appears difficult al fit^,
it may be resolved even without algebra. We have only to try
to do it inversely. Since the players, when they left off, hod
each 24 louis, and, in the third game, the first and the second
doidbled the nK>ney^ they must have bad before that last game $
Chap. 4» Of BqiMMons. 181
The 1. 12, the 11. 13, and the IIL 48.
In the second game the first and the third donUed flieir
ndnejr ; so that before that game thdy had ;
The I. 6, the IL 48, and the III. 24.
Lastly, in the first game, the second and the third gained
each aa much money as they began with ; so that at first the
tbree persons had i
I. 39, II. 21, IIL 12.
The same result as we obtained by the former solution.
536. ^u€sH(m y. Two persons owe 29 pistoles ; they haf6
both money, but neither of them enough to enable him, singly to
discharge this common debt : the first debtor sa>s therefore to
tbe second, if yon £^ve me | <rf your money, I singly will imme-
diately pay tbe debt. The second answers, tiiat he also could
discharge the debt, if tbe other would gi?e him j of his money.
Required, how many pistoles each had ?
Suppose that the first has x pistoles, and that the isecond has
y pistoles.
We shall first have^ or + 1 y ^ 29 ;
Uien aho, y .f I X =s 29.
The first efoation give5«3=29 — f y, and tbe second, a?=s
1" ^ ; so that 29 — | y = ^— ^- From this equation,
o S
V® get y=: 14| ; wherefore a?= 194*
. Smgwtr. The first debtor had 19| pistoles, and the second bad
14| pistdes.
537. '^lisfion VI. Three brothers bought a vineyard for a
hundred louis. The yougest says, that he could pay for it
alone, if the second gave him half the money which he had ; the
second says, that if the eldest would give him only the third of
his money, he could pay for the vineyard singly ^ lastly, the
eldest asks only a fourth part of the money of the youngest, to
pay for the vineyard himself. How much money had each ?
Suppose the first had x louis ; the second, y louis ; the third,
% krais I we shall then have the three following equations ;
I* a^+iyslOO« IL y + 4»a^lOO.
UL » +i « s 100 ; two of which only give tbe value of x.
182 JOgebra. Sect 4.
namclyt La:=:100 — iy, III. a? = 400 — 43^.^ So that we
have the equation,
100 — ^1^ = 400— 4x1, or 4% — ^ y = 300, which must be
combined with the second, in order to determine y and »» Now
the second equation wasy+f »=100; we therefore deduce
from it y=: 100 — ^%; and the equation found last being 4 «
— ly= 300, we have y = 8 » — 600. Consequently the final
equation is,
100 — |i»=8« — 600; so that 8|« = r00,or y«=700,
and % = 84. Wherefore y = 100 — 28 = 72, and'or = 64.
Answer. The youngest had 64 louis, the second had 72 louisy
and the eldest had 84 louis.
538* As, in this example, each equation contains only two
unknown quantities, we may obtain the solution required in au
easier way«
The first equation gives y = 200 — 9.x\ so that y is deter-
mined by X I and if we substitute this value in the second equa-
tion, we have 200 — 2a; + -|% = 100; wherefore -I » = 2 £P»-«
100, and » = 6 a: — 300.
8o that % is also determined by x ; and if we introduce this
value into the third equation, we obtain 6 a; — 300 + ^ a? = 100,
in which or stands alone, and which, when reduced to 25 a:—*
1600 = 0, gives x = 64. Consequently, y = 200 — 1 28 = 72,
and «; = 384 — 300 = 84.
539. We may follow the same method, when we have a greater
number of equations. Suppose, for example, that we have ia
general ;
I. u + — = n, IL or + y =z n, III, y + - = n,
IV. %+-^:=zn} or, reducing the fractions,
I. au-f a?=:an, II. ba7 + y=in, III. cy+^sscn^
IV. d% + U:=zdiU
Here, the first equation gives immediately x:sian'^aH9
and, this value being substituted in the second, we have a 6 n— .
a6tt-f y = bn; sothaty =ftn — a bii + a ftu; the substitution of
this value, in the third equation, gives 6cn — aicn-f-a&c tt-fstr =s
c n ; wherefore » = c n— ben -f-a6cn — abcu} substituting this
Ghap.4. qfEquiUiam. 1S$
in the fourth equation, we have cd n— • bcdn -f- a bcdn^^ah cdu
+u = dn. So that dn — edn + bcdn — ab cdn = — a bcdu + u,
or (abed — l)u=zabcdn — bedn + cdn — dn; whence we
- _ abcdn — bcdn + edn — rfn _ (abed — bed + cd— rf)'^
abed — I abed — 1
Consequently* we shall have,
abcdn — acdn + adn '-^an ^ (abed — aed +ad^-'a)
"" abed — 1 abed *— 1
^ abedn-^abdn + abn — bn ^ {abed — abd + ab'^ b)
*- abed ^ I "'*^ abed — 1 *
abcdn — aben + ben — en {abed — abe + ftc — c)
"^ a6cd — 1 "" abed — 1
abedn — ftcrfn -f- cdn — dn ^ (irAcd — 6cd + cd — d)
"" abed-^ 1 "" a6cd— 1
540. Queslioft YII. A ci4>tain has three compauiesy one of
Swissy another of Swabiansy and a third of Saxons. He wishes
to storm with part of these troops* and lie promises a reward of
901 crowns* on the following condition ;
That each soldier of the company* which assaults^ shall re-
ceive 1 crown* and that, the rest of the money shall he equally
distributed among the two other companies.
Now it is found* that if the Swiss make the assault* each sol-
dier of the other companies receives | of a crown ; that^ if the
Swabians assault* each of the others receives -^ of a crown;
lastly* that if the Saxons make the assault* each of the others
receives | of a crown. Required* the number of men in each
company? ^^
Let us suppose the number of Swiss y = * that of Swabians
= y* and that of Saxons = «• And let us also make x + y +%
= s, because it is easy to see* that by this* we abridge the cal-
culation considerably. If* therefore* the Swiss make the assault^
their number being ot, that of the other will be s — x; now, the
former receive 1 crown* and the latter half a crown ; so that we
shall have^
We find in the same manner, that if the Swabians make the
assault, we have,
tat Jt^lAm. Sect. 4.
And laflfljy thnU if tlw Saxow nount tlM SMWilty we hara^
Each of thefle three oqutions will enable us to determme one
of the unknown quantities x,yf%; '
For the fi rat gives or = 1 802 — $p
the second gives 2 y s 3708 — Sp
the third gives S » ac $604 -^ i^
If we now take the values of 6 o:, 6 y, and 6 », and write those
values one above the other, we shall have^
6x = 10812 — 6 s,
6y= 8109— 3 Sf
6 05=3 7908i^2Sf
and adding ; 6s:fz 26129-^ 11 s, or 17 • a 86129 ; so
that 8 = 1557 ; this is the whole number of aoMierBy by which
means we find»
0:=? 1 802 ~ 1537x7 265;
2 y =E 2703 — 1537 3= 1166^ or y s 583 ;
3 % = 3604 -~ 1587 3c 2067, or «s689»
M9wer. The company of Swiss conaista of 265 umi ; that of
Swabiana 583 1 and that of Sazona 689.
CHAPTBB V.
tff the Saolution 9f Pure qwidraHc Equatitme.
541. An equation is said to be of the second degree, when it eon-
taifM the square or the second power f^ the unknown quanHttff wilh-
out any of its higher powers. An equation, containing likewise
the third power of the unknown quantity, belongs to cubic equa-
tions, and its resolution requires fmrticular rules. There are,
therefore, only three kinds of terms in an equation of the sec-
ond degree.
1. The terms in which the unknown quantity is not found at
all, or which are composed only of known -ftumhera.
2. The terras in which we find only the first power of the
unknown quantity.
3. The terms which contain the square, or the second power
of the unknown quantity.
Chap. 5. OfBquMtms,. 185
80 that X i^gnirying an unknown quantity, and the letters a,
bf Cf df &c* repreaenting known numbers, the terms of the
first kind will have the form a, the terms of the second kind
will have the form b x, and the terms of the third kind wUl have
the form e 0? OP.
542. We have already seen, how two or more terms of the
same kind may be united together, and considered as a single
term.
For example, we may consider the formula axX'^hxx +
exo? as afifingle term, representing it thus (a'-^b + c) xx^
since, in fact, (a — 6 -f c) is a known quantity.
And also, when such terms are found on both sides of the
sign =, we have seen how they may be brought to one side, and
then reduced to a single teriii. Let us take, for example, the
equation,
StxX'^Sx + 4ss5xx — 8(r-fll;
Ife first subtract ^xx^ and there remains
— 3« + 4 = ^^^ — 8cr-|-ll^
then adding 8 op, we ofitain,
5x + 4zsSxx + ll;
LasUy, subtracting 11, there remains dcrd? = 5^ — 7.
543. We may also bring all the terms to one side of the sign
s=, so as to leave only on the other. It must he remembered,
however, that when terms are transposed from oHe side to the
other^ their signs must be changed.*
Thus, the above equation ^ill assume this form, 3 a; a? — 5 x -|.
7 = 0; and, for this reason also, the fMaxvifig general formida
represents all equations of the second degree*
axx ± bxzhc = Of
in which the sign zk is read plus or minus, and indicates that
such terms may be sometimes positive and sometimes negative.
544. Whatever be the original form of a quadratic equation,
it may always be reduced to this formula of three terms. If wo
have, for example, the equation
ax + d ex +/
c x + d^ gx +h*
* That if 9 the quantity thtu transposed is added to or subtracted from Cidi .
side of the equation.
Ed. Alg. 9A
186 Jttgdmu Sect. 4.
ure mtttty firef, reduce the fradtom ; mattiplyin^f for this p«r-
pose, by cx+d, we have ax + o-=^ — -r-j i— *
then hj gx + hj we have a;jN^+6;j? + aAx+6A = cex4^4.
c/x + edx +fd9 which is an equation of the second d^gree^ and
reducible to the three following terms, which we shall transpoaa
by arranging them in the usual manner :
O^agxx + hgx+thf
--^eexx+ahx — fd,
^cfx.
We may exhibit this equation also in the following fonUf
which is still more clear :
Ozz:(ag — ce)xx + (hg + ah — c/— ed)x + b h — fd.
545. Equations of the second degree^ in which all the three
kinds of terms are found, are called compleUf and the resolution
of them is attended with greater diflSiculties ; for which reason
we shall first consider those, in which one of the terms is wanting.
Now, if the term x x were not found in the equation, it would
not be a quadratic, but would belong to those of which we have
already treated. J^the ternif which contains only known numberSf
were wantingf the equation would have thisformf axxdbbx = 0,
which being divisible by x, may be reduced /o a x db b = 0, whidi
is Ukewise a simple equatiotif and belongs not to the present dass.
546. But when the middle termf which contains the first power
of x,is wanting f the equation assumes-thisform, axxztcrzO, or
a X X = =F c ; as the sign of c may be either positive or negative.
We shall call such an equation a pure equation of the second
degree, since the resolution of it is attended with no difficulty ; for
we have only to divide by a, whicl^ gives xx= — ; and taking the
square root of both sides, we find x = \ ^ / by means of wMch the
equation is resolved.
547. But there are three cases to be considered here. In tlia
first, when > is a square number (of which we can therefore
really assign the root) we obtain for the value ojna raitumol
Chap. 5. OfBquMoM. 187
imwbeff mohkh may he either inUger orfraciionaL For example^
the equation a; 2^ =144 gives a?=12« And xx^^^ gives ^
The second variety is when — is not a sgimre, in which case
ft
we iititfMherefore he comtentiBd with therign \/. If, for exaniplei
jrd?=129 we havex = vi3f the value of which may be deter-
mined by approximation, as we have already shown.
The third case is that in which — hecomes a negative number;
then the value qf xis altogether impossible and imaginary ; and this
resvU proves that the question, which leads to such an equation, is
in itself impossiMe.
548k We shall also observe before proceeding further, that
whenever it is required to extract the square I'oot of a number,
that rooty as we have already remarked, has always two values,
the one positive and the other negative. Suppose we have the
equation x x = 49, the value ofn will be not only + 7, but (Uso — 7,
which is expressed frjf x = =b 7. So that all those questions ad-
mit of a double a answer; but it will be easily perceived that in
several cases, in those, for example, which relate to a certain
number of men, the negative value cannot exist.
549. In such equations, also, as a a? j? = 6 a;, where the known
quantity c is wanting, there may be two values of x, though we
find only one if we divide by a?. In the equation xx=:.5x, for
example, in which it is required to assign such a value of a?, that
XX may become equal to 3 x, this is done by supposing a? = 3,
a value which is found by dividing the equation by a; ; but
beside this value, there is also another, which is equally satis-
factory, namely or = ; for tlien xxz=zO, and 3 a: = 0» Equa*
tionSf therefore, tf the second degree, in general, admit of two solu-
iions, whibt si^iple equations admit only qf one.
We shall now illustrate, by some examples, what we have
said with regard to pure equations of the second degree.
550. Question h Required a number, the half of which mul-
tiplied by the third may produce 24. •
Let this number =zx; ^x, multiplied by | x, must gire S4 ^
wo shall therefore have tbe equation ^xxzs:24.
188 JtigtbriL Seet 4.
Muttipljing by 6» we have xo; = 144 ; aii4 the extractfam of
the root gires a;=s d: IS. We put ds ; for ir a? aa 4. ld« we
have I OP = 6| and ^ 2^ = 4 ; now the product of these two num-
bers 1824; and if 0? = — 12» v^'e have | a; s= -^ 6, attd|d; = — -4,
the product of which is lil^ewise 24.
551. ^utslwnW. Required a number snch^ thatbyanMing
5 to it» and snbtractini; 5 from It^ the product of the mum by tho
difference would ba96«
Let this number be Xj then x + 5, multiplied by a? --« s/vwst
give 96 ; whence results the equation^ xx — 25 = 96.
Adding 25^ we havea7jr= 121 ; and extracting tin root» wo
havea;=:l]. Thus dp + 5 = 16y also ««— 5 = 6; and lastly^
6x16 = 96.
552. Question III. Required a number such» that by adding
it to IO9 and subtracting it from lOf the SQm> multiplied by the
remaindcrf ordiffcrence^ will give 51.
Let or be this number; 10+ a;, multiplied by 10-— «, mest
make 51, so that 100 — a;a: = 51. Adding xXf and subtracting
51, vc have a? j? = 49f the square root of which gives x^z7.
553. Question lY. TIfree persons, who had been piaying^
leave off; the first, with as many times 7 crowns* as tlie second
has three crowns; and the second, with as many times \7
crowns, as the third has 5 crowns. Further, if we multiply the
money of the first by the money of the second, and the money
of the second by the money of the third, and lastly, the money
of the third by that of the firs^ the sum of these three products
will be d8dO|. How much money has each ?
Suppose that the first player h&s x crawns; and since be has
as many times 7 crowns, as the second has 3 crowns, we know
that his money is to that of the second, in the ratio irfT : 3.
We shsdl therefore make 7 : 3 = or^ to the money of tke
second player, which is therefore \ x. «
Further, as the money of the second player is to that of the
third in the ratio of 17 : 5, we shall say, 17 : 5 =: | x to the
money of the third player, or to ^y^ x.
Multiplying x, or the money of the first player, by f X9 the
money of the second, we have the product ^ x a?. Then f op, the
money of the second, muItipTycd by tiie money of the tUrd> or
Chap. 5. qfEqwOum. *189
by tV^^ Ktip«8 ^^xos. Lastly f the noney of the thirdt or
'^^ X nmltiplied by x^ or the anoney of the firsts gi ve8 ^^ x x.
The sani of these three products is | d? or + ^^j ^x + -f-j-^xx ;
and) reducing these fractions to the same denominatory we find
their sum 477 ^ ^» which must be equal to the number dddOf •
We have, therefore, m x dc s SdSOf •
So that VtV^ xszl 1492, and IS^lxx being equal to 9572836,
dividing by 1521, we have a? a? = *VtiV* ^ ^^^ taking its root,
wie An&x = 'f ^^. This fraction is reducibie to lower terms if
we divide by IS ; so that a; = *|* =: 79^ ; and tience we con-
clude, that ^ ^ s 34, and y^r ^ = 1^«
Jnswer. The first player has 79^ crownSf the second has 34
crowns, and the third 10 crowns.
RiBmark. This calculation may be performed in an easier
manner ,* namely, by taking Hbe factors of the numbers which
present themselves, and attending chiefly to the squares of those
factors.
It is evident, that 507 = 3 x 169, and that 169 is the square
of 13 ; then, that 833 = 7 x 119, and 119 = 7 X 17. Now we
3 X )69
have rz — T^ xx=i SBSOf, and if wo multiply by 3, we have
9 X 169
v= — 77; ^XB 11492. Let us resolve this number also into its
17 X 49 >•
factors ; we readily perceive, that the first is 4, that is to say,
that 11492 = 4 X 2873 ; further, 2873 is divisible by 17; so that
2873 s: 17 X 169. Consequently our equation will assume the
9 X 169
fdllowing form ; r= — -r x x = 4 x 17 X 169, which^ divided bjr
9
169, is reduced to — — ^5 » a: =4 x 17 5 multiplying also by 17 x^
4 X 289 X 49
49^ and dividing by 9, we have xx=: g — : — , in which all
the ikctors are squares ; whence we have, without any further
calculation, the root x =: = -— = 79|, as before.
554. ^uestum T. A company of merchants aj^oint a factor
at Archangel. Each of them contributes for the trade, which
they have in view^ ten times as many crowns as there are part-
190 JMgAr^ Sect 4.
ncra* The profit of the factor is fixed at twice as many ctowbb
per cewtf as there partners. Further^ if we multiidy the -^^
part of his total gain by 2f 9 the imniber of partners will be
found. Required, what is that number 2
Let it be =0? j and since, each partner has contributed 10 a;,
the whole capital is = 10 a;a?. Now* for every hundred crowns,
the factor gains s^o?, so that with the capital of 10 a; a; his profit
will be I a; '• The ^^j^ part of this gain is y^x^ ; mnlti|dyiDg
by 2^9 or by V, we have -^^ x*, or f\j x^, and this must be
equal to the number of partners* or x.
We have, therefore, the equation ,^7 x^ s= xp or x^ =s 225 x ;
which appears, at first, to be of the third d^ree ; but as we may
divide by a?, it is reduced to the quadratic xx^ 2S5, whence
xs^l5.
Answer. There are fifteen partners, and each contributed 150
erowns.
CHAPTER VI.
Of the BesduHon of Mixt Equations of the S^oond Degree.
555. Air equation of the second degree is said to bemixi, or com-
pletef* when three kinds of terms are found in it, mmdyf that
whicKcontains the square of the unknorvn quantity, as a x x ; that,
in which the unknown quantity is found oidy of the first power, as
hx; lastly, the kind of terms whidi is composed only of known
quantities. And since we may unite two or more terms of the
same kind into one, and bring all the terms to one side of the
sign =, the general form of a mixt equation of the second de-
gree will be
flarirq=iarq=c = 0.
In this chapter, we shall show, how the value of or is derived
firom such equations. It will be seen that there are two methods
of obtaining it.
556. An equation of the kind that we are now considering
may be reduced, by division, to such a form, that the first term
may contain only the square or or of the unknown quantity or. We
* Sometimes called ibo afiectjM^
•*
-f
Chap. 6. OfEquaiums. 191
flhall leave the second term on the same side with o^ and trans-
pose the known term to the other side of the sign = • By these
means our equation will assume the form xx:^px = :±qf
in which p and q represent any known numbers, positive or
negative ; and the whole is at present reduced to determining
the true value of x. We shall begin with remarking, that if or a;
•{-px were a real square, the resolution would be attended with
no difficulty, because it would only be required to take the square
root of both sides*
557. But it is evident that x^-f. pa; cannot be a square;
since we have already seen, that if a root amsisU of two ternup
for example, x + Ji, its square always contatns three terms,
fuimeiy, twice the product of flu two partSf besides the square of each
part ; that is to say, the square (fx + nuxx + 2nx + nn. Now
we have already on one side xx + px; we may, therefore, con-
sider xxasthe square of thefrst part of the root, and in this case
p X must represent twice the prodtu^ of x, thefir^t part of the rootf
by the second part ; consequently, this second part must be ^ p, anil
infactthe square of x +^ p, is found tobex z + px -f.| pp.
558. Mno xi^ + px + ^p pbeing a real square, which hasf or its
roo^x-f ^p, if we resume our equation xx+px=q, we have
onlyto add^pftoboth sides, whidi gives us xx + px+^pp^q
+ ^ ppf the first side being actually a square, and the other contain-
ing only known quantities. If, therefore, we take the square root
of both sides, we find x + i P= vCTpT+q) > «•"* subtracting | p,
we obtain x=r — | p + VcTpF-Fq) 5 ^^» ^ ^^y square root
may be taken either t{ffirmativdy or neguUvdy, we shall have for x
two values expressed thus ;
=-iP=tJI
pp+q-
559. This formula contains the rule by which all quadratic
equations may be resolved^ and it will be proper to commit it to
memory, that it may not be necessary to repeat, every tin)e, the
whole operation which we have gone through. We may always
arrange the equation, in such a manner, that the pure square
xxmkj be found on one side, and the above equation have the
form X x+ px^zq, where we see immediately that
<^ = — ^P±^^pp + q.
19£ JtgAm. SeGt4«
560, The p;eiieral rale, therefore, which we deduce fipMi tfait^
in order to resolve the eqaation x x^-^p x + q^is founded oa
this consideration :
That the unknown quantity x is equal to half the coefficient
or multiplier of j? on the other side of the equation, jrfus or miniiA
the square root of the square of this number, and the known
quantity which forms the third term of the equation.
Thus if we had the equation or a; = 6 xr -f 7, we should imoie*
diately say, that x=S zh \/9T7 = 3 ± 4, whence we have
these two vaiues of or, I. or = 7 ; II. ^ s — 1. In the same
manner^ the equation j7 or =10 or — 9, would give a?=:5±
x/if^ =: 5 db 4, that is to say, the two values of x are 9 and U
56f. This rule will be still better understood, by distinguish^
ing the following cases. I. when p Is an even number ; II.
when p is an odd number; and III. when p is afractioiial
number.
I. Let p be an even number, and the equation such, that x x
ssSipx + q; we shall, in this case, have x:=ipdt \/pp^q.
II. Let p be au odd number, and the equation xxzspjc + q;
we shall here have or = I p =h l—pp + qy and since ^pp + fac:
^^^^^ — 't we may extract the square root of the denominator^ and
write ar = |p±i^:^^tii=P±yiH±lI
III. Lastly, if p be a fraction, the equation may be resolved
in the following manner ; let the equation he axx:=hx + CfOT
h x^ c 6
xxz= — ^ + — , and we shall have by the rule, x =
a ' a' '" ' 2a
4
bb c -.^ hb e bb + 4ae.. ^ ,. ^
+ -. Now, -r — + — = --z f the denominator of
4aaa 4aa'a 4aa
which is a square ; so that x = ^=^ V^^-«-4ac^
562. The other method of resolving mixt quadratic equations,
is to transform them into pure equations. This is done by sub*
stitution ; for example, in the equation xx^px+qf instead of
the unknown quantity a?, we may write another unknown quan-
tity y, such, that or = y + ^ p ; by wliich means, when we have
determined y, we may immediately find the value of x.
as we
Chap. 6. Of Equations. . 193
If we make tins substitution o^y +^p instead of ar» we have
xx^yy+py-^^ppf and p^=Pl/ + -J-pp; consequently our
equation will become 1/ f/+py +iPP = py + iPP + 99 which is
first reduced, by subtracting py. toyi/ + ^pp = ^pp + g; and
then, by subtracting ^pp, toy y=lpp + 5. This is a pure
quadratic equation, which immediately gives y = d= 1-^ pp + g.
Now, since a: = y + ^ p, we have ^=^^p:^ A-pp + qf
found it before. We have only, therefore, to illustrate this rule
by sonte examples.
563. Quesn m I. There are two numbers ; one exceeds the
other by 6, and their product is 91- What are those numbers ?
If the less is a*, the other is or + 6, and their product x a; -f 6 a:
= 91. Subrrac tinj^ 6 ar, there remains a:a- = 91 — 6x, and the
rule gives a; = — 3 ± v'g+Ql^ — 3 ± 10 ; so that x=z7, and
flc = — 15.
Answer. The question admits of two solutions ;
By one, the less number a; is = 7, and the greater rt -f 6 = 1 3.
By the other, the less number x =z — 15, and the greater
« + 6 = — 7.
564. ^lestion IL To find a number such, that if 9 be taken
from its square, the remainder may be a number, as many units
greater than 100, as the number sought is less than 23.
Let the number sought = x ; we Icnow, that xx — 9 exceeds
100 by a* a: — 109. And since x is less t^an £3 by 23 — x, wo
have this equation 5 xx — 109 = 23 — x.
Wherefore xx=z — x+ 132, and, by the rule.
So that X = 1 19 and x = -^ 12.
Jnswer. When only a positive number is required, that
number will be. 11, the square of which minus 9 is 112, and
consequently greater than 100 by 12, in the same manner as 11
is less than 23 hy 12.
565» ^slion III. To find a number such, that if we multi- "
ply its half by its third, and to the product add half the number
required, the result will be 30.
£11^ Mg. 25
»
}94 Jllgdbra. Sect. 4.
Suppose that namber = x, its half, muKiplied by Hb tliird,
wUl make ^xx; so that } a: a: + 1 x = 30, Multiplying^ by 6,
we haye xx + 3a:=i80, orapa; = — Sx + 180, which gives
Consequently x is cither = 12, op — 15,
566. ^sfion IV. To find t^'o numbers in a doable ratio fo
each other, and such that if we add their sum to their product,
we may obtain 90.
liOtoneof the numbers =:X, then the other will berzSx;
their product also z^Zxx, and if we add to this 3 x, or their
sum, the new sum ought to make 90. So that ^xx-^-SxssdO;
2070? = 90 — 3x; xxzs — I X + 45, whence we obtain
|9 3 27
Consequently op = 6, or — 7|.
567. ^esium Y. A horse dealer, who bought a borse for a
certain number of crowns, sells it again for 119 crowns, and his
profit is as much per cent, as the horse cost him. Required,
wiMit he gave for it ?
Suppose the horse cost x crowns ; then as the bone dealw
gidns X per cent, we shall say, if 100 give tlie proAt » ; what
does X give ? Answer, — . Since, therefore, he has gained — ,
imd the horse originally cost him x crowns, he niqst ba%'e sold
it for 0? + —J 5 wherefore x + — = 119. Subtracting x,
wehave — =— 0P + 119J and multiplying by 100, we have
apo?aB—100ar-f 11900. Applying the rule, we fiiida;=:~
50 it V2500 + 11900 = — 50 db \nim = — 50 zfc 120.
Answer. The horse cost 70 crowns, and since the horse
dealer gained 70 per cent, when he sold it again, the profit must
have been 49 crowns. The horse must have been, therefore,
sold again for 70 +49, that is to say, for 1 19 crowns.
. 568. (ly^sUon YI. A person buys a certain number of pieces
of cloth ; he pays, for the first, 2 crowns ; for the second, 4
crowns ; for the thirds 6 crowns^ and in th^ same manner always
St crowm nun^ fcr «ftfb Mtoiewg picice* N6W, sll Ihii j^w
together cost him 110. How many pieces had he ?
Let the iMimber sought = x. By the qiMtion the purebaser
paid for the diflRsrent pieces of cloth in the following manner ;
for the 1, £» $9 4, 5. • • • OP
he pays 2» 4» 6, 8y 10 . • . 4 d a? crowns*
It la therefMTB required to find the sum of the arithMetical
progression 2 + 4+64.8 + 10 -f S a?, which consists of
X termsy that we may deduce from it the price of all the pieces
•f dotti token togrtker. Thenrie which we have already gifen
for this oprration^ requires us to add the last term and the first ;
the sum of which is d x + 2 ; if we multiply this simi by the
■umber of terms Xf the product will be fi x x + S x ; if we lastly
divide by the difference 2, the quotient will be ^ jr + o^f which
la the sum of the progression $ so that we have xx + x^ 1 10 ;
irherrfbre ar 4? :i5 — ^ + 1 1 0>
and x = ^^f — ^ 110=^- — I =
10.
Answttm Tlie number of pieces of cloth is 10.
569. (pustion YII. A person bought several pieces of clotb^
ii»r 180 crov^ns. If he b%d received for the same wm S pieces
more, he would have paid three crowns less for each piece ;
Bow many pieces did he buy ?
Let us make the number sought = x; then each piece wiH
180
have cost him — crowns. Now, if the purchaser had had or + 3
X
pieces for 180 crowns, each piece would hare cost r cpowna ;
and, since this price is less than the real price by three crowns^
we have this equation,
180 _ 180
•V + 3 X
180 x
Multiplying by a?, we have — — ■ = 180 -^ 3 ^ ; dlTiding by
60 .T
3. we have ' = 60 — * ; multiplying by a? + 3 we have
Wx:al90 + 5Tx-^xX} addlng^or, we shall have a; ^ + 60^'
%
196 JKgAm* Ssct 4.
s 180 + 57 « ; subtractu^ 60a?, we sbdi *•▼« xjg s — 5 or 4.
180.
The rule> conseque ntly gives
«dnst(7er. He bought for 180 crowns 12 pieces of rioth at 15
crowns Ihe piece» and if he bad got 3 pieces more, nannely 15
pieces-ior hSO crowns* each piece would hare cost only 19
crowns, that is to say, 3 crowns less.
570. ^estion YIH. Two merchants enter into partnersbip
with a stock of 100 crowns ; one leaves his money in the part-
nership Cor three months, the other leaves his for two months*
and each takes out 99 crowns of capital and profit What pro-
portion of the stock did each furnish i
Suppose the fii'st partner contributed x crowns, the other will
have contributed 100 — or. Now, the former receiving 99
crowns, his profit is. 99 — ,r, which he has gained in three
months with the principal jp ; and since the second receives also
99 crowns, bis profit is or — 1, which he h^s gained in two
months with the principal 100 — x; it is evident also, that the
3 x-«3
profit of this second partner would hiMre been — - — , if he had
remained three months in the partnership. Now, as the profits
gained in the same time are in proportion to the principals, we
3 X — 3
have the following proportion, 07:99^— a: = 100 — x z •
The equality of the product of the extremes to that of tho
means, gives the equation.
So? J? — 5a?
=r 9900— -199 or -(.a?x;
Multiplying by 2, we have Sxx — 3 0? = 19800 — 398 x
+ 2 0? a? ; subtracting 2 a? a?> we have xx — 3 a?= 19800 — 398 a?
adding 3 or, we have xxsz 19800 — 395 x.
Wherefore by the rule,
395
x^ 5.
|l 56025 79200 [395 485 90
• Answer* The first partner contributed 45 crowns, and the
other 55 crowns. The first, having gained 54 crowns in three
Bioiijths, would have gained in one inonth 18 crowns; and the
second having gained 44 crowns in two months, would have
gained 22 crowns in one month : now these proiits agree ; for,
if with 45 crowns 18 crowns are gained in one month, sa
crowns will be gained in the same time wilh 55 crowns.
571. ^tstion IX. Two girls carry 100 eggs to market $ one
bad more than the other, and yet the sum which they both re-
ceived for them was the same. The first says to the second, if
I had had your eggs, 1 should have received 15 sous. The
other answers, if I bad had yours, I should have received 6|
sous. How many eggs did each carry to market ?
Suppose the first had x eggs ; then the second must have had
100 — X.
Since therefore the former would have sold 100 — ^or eggs for
15 sous, we have the following proportion ;
100 — X : 15 = or • • . « to --— sous.
100 -«-x
Also, since the second would have sold x eggs for 6^ sous, we
find how much she got for 100 — jt eggs, by saving
20 ^^^ . 2000— Oar
x: -z-^ 100 — ar....to .
3 3x
Now both the girls received the same money ; we have con-
\5 X 2000 Sn "V*
■equently the equation, ^q^_^ = — , which becomes
this,
Q5xx:= 200000 -— 4000 x ;
and lastly this,
xxsz — 160ar-f8000;
whence we obtain
x= — 80 + V6400+ 8000 = — 80 + 120= 40.
Aiswer. The first girl had 40 eggs, the second had 60, and
each received 10 sous.
572. qustion X. Two merchants sell each a certain quantity
of stuff; the second sells 3 ells more than the first, and they
received together 55 crowns. The first says to the second, I
should have got 24 crowns for your stuff; the other answers,
and I should have got for yours 12 crowns and a half. How
" many ells had each ?
Suppose the first had or ells; then the second must have had
I9t Jig^rm. fertL4x
524 crowns, be must have recdTcd — -r-r crowns for Ua jr db»
And likW rrgmrd la iha aecoad^ ance ba wdiiM have lald « db
for 15^1 crowas, he raosf have sold his x -|- Sefls far ^^ •
ao that the whole sum ther receirei was — r-i + — -^ — = SS
crowns.
IfaiA equation becomea drx = 20jr*^75f wbeace we lian
flp= 10 db vlvA* — r5 = 10 ±5.
J^nswer* The question admits of two solutions ; accordiiig to
the first, the first merchant had 15 ells, and the second bad 18;
and since the former would have sold 18 elb for 24 crowns» be
must have sold his 15 ells lor 20 crowns ; the second, wbo waaM
have sold 15 ells for 12 crowns and a half, most have sold bis
18 ells for 15 crowns ; ao tbat thej actual! j received 55 crowns
for their commodity.
According to the second solutiot^ the first aierrbant bad S
elb, and the other 8 ells ; so that, since the first would have
sold 8 ells for 24 crowns, he mciSC have received 15 crowns for
bis 5 ells i and since the secoad wouM have sold 5 ells for 12
crowns and a half, bis 8 db must have produced him 20 crowns.
The sum is, as before, 35 crowns^
GBAPTER Vlf.
Of Oie Jftiture of Sqnattms Uf Iht Sfami Degrtt.
573. What we have already said sufficiently shows, ^tbat
equations of the second degree admit of two sedations ; and this
propertjr ought to be examined in every point of view, because
the nature of equations of a higher degree will be very muck
illustrated by such an examinati.in* We shall therefore retrace^
with more attention, the reasons which render an equation of
the second degree capable of a douGle solution ; since ttiey un«
doubtcdiy will Qxbi|>it an essential property of those equatioaa.
Chap. 7. Of EtitaUens. IM
574» We have already seen, it is true, that this double sola-
tiuQ arises from, the circumstame that the square root of any
number may be taken either positivelyf or negatively ; however^
as this principle will not easily apply to equations of higher
degrees, it may be proper to illustrate it by a distinct analysis*
Taking, for an example, the quadratic equation, jr x =s 12 a? — S5p
we shall give a new reason for this equation being resolvible in
two ways, by admitting for a? tlie values 5 and 7^ both of which
satisfy the terms of the equation*
575. For this purpose it is most convenient to begin with
transposing the terms of the equation, so that one of the sides
may become 0; this equation coasequently takes the form a:j?
— 12 or 4. 35 = 0; and it is now required to find a number
quch, that, if we substitute it for x, the quantity arx^^^i^x + SS
may be really equal to nothing ; after this, we shall have to
show how this may be doi^e in two ways.
576. Now, the whole of this consists in showing clearly, that
a quaniitff of the form xx— 12x + 35 may be anrndered as the
product of two factors; thus, in fact, tlie quantity of which we
speak is composed of the two factocs (x — 5) x (a: — 7). For,
since this quantity miist become 0, we must also have the pro-
duct (a? — 5) X (*r — 7) s= ; but a product, if whatever number
(f factors it is composed, becomes = 0, only when one of those faC'
tors is reduced too,* this is a fundamental .principle to which we
must pay particular attention, especially when equations of sev*
oral degrees are treated of*
577. It is therefore easily understood, that the product (x *-^ 5)
X (x — 7) may become in two ways : one, when the first fadar
X — 5 = ; the other, wlan the secondfadar x — 7 = 0. In tim
first case yiz^5,in the other, x^7. The reason is, tlierefiDre^
very evident, why such an equation or a? — 12 or .{. 35 = 0, ad-
mits of two solutions, that is to say, why we can assign two
values of or, both of which equally satisfy the terms of the equ»»
tion. This fundamental principle consists in this, that the
quantity xx — 12x + S5 may be represented by the product oC
two factors.
578. The same circumstafices "^are found in iUl eqoationsof
the second degree. Fori after Jbaidog brought j(ll 4be tenna te
900 Mgebra. Sect. 4.
one side, we always find an equation of the following form jcjp
*— ax + 6 = 09 and this formula may be always considered as ^
the product of two factors, which we shall represent by (x — p)
X (.r — q), without concerning ourselves what numbers the
letters p and'q represent. Now, as this pi*oduct must be = O5
fk*om the nature of our equation it is evident that this may hap-
pen in two ways ; in the first place, whenx = p^ and in the
second place, when x = 9 ; and these are the two values of x
which satisfy the terms of the equation.
579. Let us now consider the nature of these two factors, in
order that the multiplication of the one by the other may exactly
produce X X —^ a X -f &• By actually multiplying them, we get
XX — (p +q) X -f p q ; now this quantity must be the same as
XX — a X -f fr, wherefore we have evidently p -f 9 = a, and p (
=:fr. So that we have deduced this very remarkable property,
that in everif equation of the farm xx — ax-fb = 09 the two
values of x are sudi^ that their sum is equal to a, and their product
equal toh; whence itJaU^ncs that, if we know one rf the valueSf
the other also is easily found.
580. We have considered the case in whic h the two values
of X are positive, and which requires the second term of the
equation to have the sign — , and the third term to have the
sign +. Let us also consider the cases in which either one or
both values of x become negative. The first takes place when
the two factors of the equation give a product of this form
fx — p) X (Jp + ?) 5 for then the two values of x are x =p, and
X = •— 9 ; the equation itself becomes x x -f (9 — p) x — p f = ;
the second term has the sign -f , when q is greater than p, and
the sign — , when q is less than p ; lastly, the third term is
always negative.
The second case, in which both values of x are negative,
occurs, when the two factors are (x +p) X (x + q); for we
shall then have x = — p and x = — g • the equation itself be-
eoroes a?x + (p + 9) X +p 9 = 0, in which both the second and
third terms are affected by the sign -f.
58 1 • The signs of the second and the third term consequent! j
show us the nature of the roots of any equation of the second
degree. Let the equation be xx. . ..ax,... b=:0, if the
Chap. 7. OfRfiuamif. aoi
second and (bird terms bave the sign +9 the two values of x are
both negative ; if the second term has the sign ^-^9 and the third
tirm has -f, both values are positive ; lastly, if tlie third term
also has the sign —9 one of the values in question is positive*
But in all cases, whatever, the second term contains the sum
of the two values, and the third-term contains their product*
582. After what has been said, it will be very easy to form
equations of the second degree containing any two given values
Let there b^ required, for example, an equation such, that one
of the values of x may be 7, and the other — 3. We first form
the simple equations x=7 and a; = -— 3 ; thence these, x — 7
= and 0? + 3 = 0, which gives us, in this manner, the factors of
the equation required, which consequently becomes or a? — 4 or—
21 = 0« Applying here, also, the above rule, we find the two
given values of a;; for if a? a; = 4 a? + 21, we havex=s2 dz ^25
= 2 =b 5, that is to say, x = T, or or = — 3.
583. The values of x may also happen to be equal. Let there
be sought, for example, an equation, in which both values may
be = 5« The two factors will be (a? — 5) x{x — 5), and the
equation sought will be a: a: — 10 a? + 25 =: 0. In this equation^
X appears to have only one value ; but it is because x is twice
found = 5, as the common method of resolution shows; for we
have xx=\Ox — 25 ; wherefore a:= 5 db ^^0"= 5 d= 0, that
is to say, x is in two ways = 5.
* 584. A very remarkable case, in which both values of x be-
eoipe imaginary, or impossible, sometimes occurs ; and it is
then wholly impossible to assign any value for a?, that would
satisfy the terms of the equation. Let it be proposed, for ex-
ample, to divide the number 10 into two parts, such, that their
product may be 30. If we call one of those parts x, the other
will be = 10 — x, and their product will be lOx — a?a; = 30;
wherefore xx=^\Ox — 30, and Xz:z5zt\/I^^ which being
an imaginary nunAer^ slurws that the question is impossiUe^
.585. It is very important, therefore, to discover some sign,
by means of which we may immediately know, whether an
equation of the second degree is possible or not.
Let us resume the general equation a a? *— « a; ar + ft = 0.
BtiL JUg. 26
SOS JifgAra. SecL 4.
-=-j-J-:
or
We shall have xx = ax — b, and * = -gadzJ-aa — J.
This shows* that if bis greater than }aa, or4 b greater tbaii a o^
the two values of jt are always imagioary^ sioce it would be
reqaired to extract the square root of a negative q^iMiti^ ; oa
the contrary, if ftisless than^ao* or even less than Oythitf is to
say* is a negative number* both values will be possible or real.
JBut whether they be real or imaginary^ it is no leas true* that
they are still expressiUef and always have this piwper|y» Ibat
their sura is =a* and their product ^ b. In the eqmtion X9
•^ 6 a; 4- 10 s= 0* for example, the sum of the two valuea of « mast
be =? 6» and the product of these two values most be = 10; now
we find* I. a;^S4-v^ wd 11. «=9;3-^v>n'« qaantitiei
whose sum s= 6^ and the product = 10.
586. The expression* which we have just fonnd, mf be
represented in a manner more general* and so aa to be ap|4ied
to equations of thisform]/;v^d:f « -l-AsO; for this equation
X = — s. — ^r^^^ff ^ . whence we conclude that flie two Tallies
9Fe imaginary* and consequently the equation impossible, when
4/ A is greater than gg; that is to say* when* in the equation
fxx — gx + h = 0* four times the product of the first and flm
last term exceeds tho squall of the second term : for the product
of the first and the last term* taken four times* is Ajhxx^ and
tbe square of the middle term is g go? x; now, if 4/Ar a? is greater
than ggxXf 4/ A is also greater than gg* and in that caae^ tho
equation is evidently impossible. In all other cases the equa-
tion is possible* and two real values of x may be asaiguad.
It is true they are often irrational ; but we have already aeaut
that* in such cases* we may always find them by approxima-
tion ; whereas no approximations can take ^ace with regard to
iolaginary expressions* such asyC^J; for 100 is as farfroBi
being the value of that root* as 1* or any other ipinber.
58r. We have further to observe, that oay fiumti^ if tte
second di^ee* X X zb a X di by mta< oApay^ be rtsdxiMt into tw9
factors, such as (« =b j?) x (« ± 4)7**''*afcjt we twk three
V
Chap. 7«
CfBquatums^
203
factors^ such as theses we should come to a quantity of the third
degree, and taking only one such factor, we should not exceed
the first degree.
It !s therefore certain that eroery equation of the second degree
fueessarily contains two values of x, and that it ean neither harce
more nor less.
m
588. We have already Seen, that when the two factors are
found, the two values of x are also known, since each factor
gives one of those values, when it is supposed to be = 0. The
converse also is true, vi». that when we have found one value of
X, WO' know also one of the factors of the equation ; tor \( xezp
represents one of the values of x, in any equation of the second
degite, jv •— p is one of the factors of that equation ; that is to
say, all the terms having been brought to one side, the equation
is divisible by iv-^p; and farther, the quotient expresses the
other factor.
589. In order to illustrate what we have now said, let there
be given the equation 2?^+ 4 j? — SI =70, in which we know
that a; = S is one of the F ah w s <rf x, because "sTs^ -f TxT)
»-i£l=:D; this shows, that a?— '3 is one of the factors of the
equation, or that jv^ +4 or — si is divisible by a? — S, which
the actual division proves.
a?-**$)xaf + 4 jp — SI (x + 7
xx-^Sa:
7dr — SI
Tor — Si
0.
So that the other ftactor is x + 7, and our equation is i«pi«.
sented by the product (^—- 3) x (^ + 7) & ; whence the two
values of x immediately follow, die first factor giving « =s 3,
and the oflier or = — 7.
;
^
/,
f
/ Ol i
^* t^
m
MM
ftUESTIONS FOB PRACTICE.
Fractions,
SECTIOir I. CHAFTEB 9.
1. Redace -rr ^^^ -- to a common denominator.
a c
Jns. and — .
ac a c
2. Reduce -r and to a common denominator.
c
. «c J ab + b*
•attSm -7 — and — . •
be be
S X S ft
3. Reduce -^9 —f and d to firactioils having a common de-
nominator. •«««• ^rrrf m *Da -^ .
o a e oae oa c
4f Reduce -r^ --r- and a -| to a common denominator.
^ 9a Bax , 12a« +S4x
• 12 a' 12 a' 12a
5. Reduce -rrf -r • and • — ■. — to a common denominator.
2 3 X + a
Sx + 3g 2 tt» X + 2fl» 6x» + 6a»
6x + 6a' 6x + ba * 6x + 6a'
6. Reduce - ^ - ^ rr-» and — t to a common denominator.
2a^6 2a » c ^ 4a^ d
•«^- -^' -T^r * a^d X^
205
SBOTIOH I. CHAPTER 10.
7. Required the product of ~ and -^. An$. ^.
«• Required the product of -, -r— , and -^r-* •*»• "itt-*
9. Required the product of — and — 7—. Aw. — =-^ .
10. Required the product of --^ and -r— . Au. "oT"-
11. Required the product of -r— and --^. Aw. -=^.
12. Required the product of -^, -^, and -^. Aw. 9 a or.
13. Required the product of 6 + -^ and — . * Aw. il±-if .
ax X
X* MM h* x^ 4- 6'
14. Required tjie product of — j and --— — .
Am.
15. Required the product of Xf , and
6«c +6c**
Jins,
x^
a^+a^
16. Required tbe quotient of ~ divided by ^. Aw. 1—.
17. Required the quotient of -^ divided by -^. Jbis. •^.
18. Required the quotient of ^^"^^. divided by -^-iJL.
Sx-— 29 "^ 5 X -^ a
2x* —2d*
2 X*
19. Required the quotient of - ^ ^ divided by ^
Aw.
x+ a*
9x
jp*— ax+a»*
20. Required the quotient of ^ divided by ^. Jhis. — .
5 13 60
21. Required the quotient of ~» dividedj>y 5,x. ,.«iw. —
< :
\
206 JMgebrtL
£2. Required the quotient of > - ^ ■ dividod ky — . •Am. — H-,
25. Required the quotient of -5— j divided by -^- Aig. ^ ,
24. Required tbe quotient of , a 6 x 4- 6« ^*^*^^d ^•y
X — b ^ jc
h^fixit Series.
8BCTION II. CHAPTER 5.
25. Resolve into an infinite series.
c— X
26. Resolve — ; — into an infinite series.
a + x
b bx bx* bx^ . «
a a* * a^ a*
or resolved into factors,
b ,^ a^ x^ x^ ^ ^
a ^ a*^ a* a' ^ ^
—J
2r. Resolve - — -r- into an infinite series.
X + b
a* h ^ b^
^n5.-x(l-- + -,--j + &C.)
1 +JP
28. Resolve ; into an infinite series.
1 X
JSns. l+2x + 2x^+2xs+^x\&c
a»
i29. Resolve , — ; — rr into an infinite series.
(« + xy
^ . Zx Sx^ 4 JT* .
^
"■■I ^—•^•1^^
■"'^
questions far Praake. 207
Sfurds or Irrational Jiiimbers,
sBCXloir I* CHAFTEBS 12, 19 ; and section n. chApteb 8, kc.
SO. Reduce 6 to the Fonn of x/T» Ms. V36.
31. Bediice a -fi to the form of vTc* •A'^* Vaaf ta6-f6&.
52. Reduce r— = to the form of vJ. JIns. V ttt-
SS. Reduce a* and ft to the commoii exponent -r.
Ans. a \^9 and h^\^.
34. Reduce V48 to its simplest form. Ms^ 4 \^T*
35 Reduce VaSx— a^xt to its simplest form.
•Aim. a v^ax*-xx«
* -
^-7 — — to its simplest form.
3
t
3a&
37. Add v^s" to 9 y'g"} and \/b to v^. ^fu. 3 v?"; and >/98«
4
33. Add x/4a and v^ together. Jlns. (a + S) v^.
66 + CC
39. Add 3» and -Jp together. .tfii. ^ ^ -
: . 4 _
40. Subtract via from va*« •^^>^* (a*- 2) y^ai
'41. Subtract ^i from |*- Jiw. ^ill^" VJ-
42. Mukiply ^|1I} by V '^. Ms. ^fi^
S 6
43. Multiply \/d by Va^* •An^* \/a^ 6'<^^.
44. Multi^y \/4a — 3^ by 2 a. .^iw. Viea* — i2a*ar.
45. »|p|tiply -^ VJ^rr by (c—d)s/r^.
Jtns* "' ok — V^'*' X — a a:*.
1
J
208 JggehiL
46 Multiply vr- VT- \/Thys/^ + \/r«- V3
47. Divide a* by a^ j and a*» by a"** djiw. tf" and a ^ ^ ,
VLn
48. Divide ^^^^* Vfl*x— ax« by -j^ Vj:ri.
49. Divide a* — ail — h+d y/jT by a — v*^
Ms, a -f v^ -— ^
50. What 18 the cabe of s/T'i M$. vST
a 3
51. What is the square of 3 v6 «* ^ «^* ^ ^ V^' « •
52. What is the fourth power of -^ Sfizi ^
Jtns.
a«
4 6*(c«— 26c + 6«y
53. What is the square of 3 +\/T 7 Jku. 14 + 6 ^T
s
54. What is the square root of a* ? Am. d^ ; or \/a3.
1 s
55, What is the cube root of a ft* ? Jins. abV^ ; or v^a 6 6.
56. What is the cube root of ^/a* — x' ? Jln«. x/a^ — xK
57. What is the cube root of a* — v«J^--*^ '
•anf. Va» — %/aa7 — xs.
58. What multiplier will render a + \/3 rational ?
Ans. a — vF-
59. What multiplier \rin render x/a — \/r rational ?.
.tftw. x/T + vF.
GO. What multiplier will render the denominator of the frac-
tion _ _ rational? ••' Ms. Vf"— vST
8ECTI0K II. CHAPTEB 12.
61 . Besolve y/a* + x* into an infinite series.
x« X* x^ ^*5^« -
^ 3o 8o' ^l6tt* 828^0^
c-f
(fiuiHmt for Fraetiee. fi<
62* Benira yTTT into n infinite series.
■** * + £ — 1+ J6~32'*
63. B«aolve \a* — x' into an infinite series.
. X* _5l_.^ J
Jtu. a — 5
64. Resolve \ l — x* into an infinite »
Cff< RflSidTe V''* — ^* '"*<* **" infiiite series.
•*'"•'"'" Tr" ~ 8H ~ 167' ■
66. Resolve — :::= into an infinite series.
T»*
tt^sto'^Sa' ^ 48a' '
67. Besolre (a* — x*)'' into an infinite series.
«- T /. ^* 2*« ear* .
*• 0* 35 a* 125 o*
. B«8olre f** '^•''^ into an infinite series.
. Resolve . f " .^ - . -rr into an infinite series.
. ! „ ix* Sx* 40jr«
£himinatuin 0f «9ri(AnKlieaI Fro^rMsimi.
SECTIOK III. CBAFTEB 4.
70. REq,TiTRED the sum of an increasing arithmetical pr
gresflion. having 3 Tor its first term, i for the common difibrenc
and the nnniber of terms SO. Ai». 440.
71. Required the sum of a decreasing arithmetical {irogrc
BuL^. 37
£10 Mgebra.
sion, having 10 for its first term, | for the comiiim diflTerencey
and the number of terms 21. Jin», 140*
72* Required the number of all the strokes of a clock in
twelve hours, that is, a complete revolution of the index.
Jhis. 78.
73. The clocks of Italy go on to 24 hours ; how many strokes
do they strike in a complete revolution of the index ? .tfiu. 300.
74. One hundred stones being placed on the jground, in a
straight line, at the distance of a yard from each other, hoW far
will a person travel who shall bring them one by one to a
basket, which is placed one yard from the first stone.
Jius. 5 miles and 1300 yards.
. The greatest Common Divisor.
SECTION III. CHAPTEB 6. — SECTION I. CHAPTES 8.
75* Reduce — =— — =— to its lowest terms. Aw, — =-.-
76. Reduce ^ . ^ -r — —rr to its lowest terms. Ms. - — r-r~ •
77. Reduce -- — rs — ; to its lowest terms. Ans. 1 — •
X* — 6*0?* x^
78. Reduce -^ — ^ to its lowest terms. Ms. — r— ^ — --.
79. Reduce -r ^ ""^ ^^ ^3 iQ^^st terms. Jins. ^ ■,
on T> J 5o« + 10a*x + 5a»J?« * ...
80. Reduce ^^ ^ + 2a^x» +Uax^ + x^ *^ '^^ '^^^«* *^™s.
Ms. ^«* + 5«''
a'* x + ax* +x**
Summation of Oeometrical Progressions.
SECTION III. CHAPTER 10.
81. A SERVANT agreed with a master to serve him eleven
years without any oth^r reward for his service than the pro-
duce of one wheat corn for the first year ; and that product to
be sown the second year, and so on from year to year till the
^estions fir Pradiee. 211
end of the time* allowing the increase to be only in a tenfold
proportion. What was the sum of the whole produce ?
klifhs. mill tllUO wheat corns.
N. B. It is further required, to reduce this number of corns
to the proper measures of capacity, and then by supposing an
average price of wheat to compute the value of the corns in
money.
82. A servant agreed with a gentleman to serve him twelve
months, provided he would give him a farthing for his first
month's service, a penny for the second, and 4d» idt the third,
&c. What did his wages amount to ? Ans. 5SQ5L 8s« S^d.
> 83. Sessun an Indiauy having invented the game of chess,
showed it to his (^iiice, who was so delighted with it, that he
promised him any reward he should ask ; upnn which Sessa
requested that he might be allowed one grain of wheat for the
first square on the chess board, two for the second, and so on,
doubling continually, to 64, the whole number of squares ; now
supposing a pint to contain 7680 of those grains, and one quar-
ter to he worth 1/. 78. 6d, it is required to compute the value of
the whole sum of grains, dns. £64481488296.
Simple Equations.
' SECTIOir IV. OHAFTER 2.
84. If x^4 + 6=:8,then willap=:6.
85. If 4a? — 8 = 3 a: + 20, then will a: = 28. '*
86. If aa! = a6 — a, then will a: = &—l.
87. If 2 a; +4 = 16, then will or = 6.
3e*
88. If a a: 4- 2 6 a = 3 c*, then will x = — •— 2 1.
89. Ij; -1= 5 + 3, then will x=: 16.
90. If ^—2 = 6 + 4, then will a:= 18.
91. If a*—— = c, then will a:= ' .
92. If 5ar— .15=:2a? + 6, then will a? = r.
93. If 40 — 6af— 16 = 120— Mar, then will»= 12.
2153
ytfgAra,
94. If -5 — - + - = 10, then will x = 24.
% u 4
95. If ^LZli + ^ = 20 — ^^^, then Will a: = f 3 J.
96. If JE a? + 5 = 7, then will x=:6.
97
. If x + Sa^+x* = -J!L— » tlien will x:=:a iL.
98. If 3iia?+4— 3 = fta?— CI, then wiH a? =:^^=4ri*
99. If V12 + X = 2 + v^ then will a: = 4.
1 2o* 1
100
— T, then will y = -^ a v/S:
101
then will y =5 1S«
102. If v^ + Va + 0? = ^ » then will a:s 4*
103. If V «' + ^* = V** + a?*i then will x = ^ J^a^ .
104. If a?= V*** +^V6« + xs — It, then will ar = 4- —a*
,^ 128 216
106. If -^^ = tr-A,f then will a? -- 8.
a? — a X — 3
lor. If
45
57
2a? + 3 4a: — 5
, then wina:s=6«
108. If — T = — - — , then win ar = 6.
3 4
109. If 615 X— 7 a;* = 4S x, then will a; = 9.
SECTION IV. OHAPTEB 3.
'' 110. To find a numher, to which, if there be added a half, a
third, and a fourth of itself, the sura will he 50. Jhu. 24«
111. A person being asked what his age was^ replied that'l
<
^
HMsRons fir Pradtee. 819
qi hta age multiplied by -^^ of his age gives a product eqnat to
bis age. What was his age ? Att, 16.
112. The sum of 66()t. was rtUaed for a particular purjmse by
four persona, A, B, C, and 0; B advanced twice aa mucli A ;
C aa much as A and B together ; and D as much as B and C.
What did each contribute 7 Jins. 6U, 120;, 18OX. and 3001.
lis. To find that number whose \ part exceeds iIh ^ part
by IS. ^>ia. 144.
114. What mimof money is thnt^ whose 4p^i^> 1 Pi"^' "'"^ t
part added together* amuuiil to 94 poumla? Jlns. liOl.
115. In a mixture of copper, tin, and lead, one half (if tfib
whole — 16/6. was copper ; 4 of the whole — lilS. tin ; and 4.
of the whole -f 4(6. lead : whatquantily of each was there in the
composition ?
.3m. liBlb. of copper, 8416. of tin, and 76lb. nf lead.
116. What number ia that, whose 4 part exceeds its -} by 72 1
Aas. 540.
117. To find two numbers in the proportion of e to 1 , so tiiat
if 4 be added to each, the two sums shall he in th« proportion of
9 to S. tSns. 8 and 4.
118. There are two numbers such that ^ of the greater added
to 4 of the leas ia I S, and if ^ of the len be taken from ^ nf the
greater* the remainder is nothing ; what are the numbers ?
^ns. 18 and IS.
119. In tiie composition of a ceKain quantity of gunpowder ^
of the whole plus 10 was nitre; i uf the whole minus 4^ was
Stt^pAtir, and the chareoai was | of the nitre — S. Bow many
pounds of gunpowder were there 7 Jim. 69,
120. A penon has a lease for 99 years ; and being asked
how much of it was already ex|>ired, answered, that two thirds
of the time past was equal to four fifths of the time to come :
required the time past Jns. 54 yeara.
121. It ii required to divide the number 48 into two such
parts, that the one part may be three times as much above SO
aa the other wants of 20. Mt. 33and 16.
122. A person rents 25 acres of land at 7 pounds 12 shillings
per annum ; tbis land couaisting of two sorts, be rents the better
i214 JBgebra.
sort at 8 sbillings per acre, and the worse at 5 : required th«
number of acres of the better sort. JBlm. 9.
123. A certain cistemy which would be filled in 12 minutes
by two pipes running into it, would be filled in 20 minutes by
one alone. Required^ in what time it would be filled by th«
other alone* ^ins. SO minutes.
124. Required two numbersy whose sum may be s, and their
proportion as a to 6. Ans. — rr ^^^ — n:«
** a+ h a + b
125. A privateer, running at the rate of 10 miles an hour,
discovers a ship 18 miles off making way at the rate of 8 miles
an hour ; it is demanded how many miles the ship can run be-
fore she will be overtaken ? Jn$. 72.
126. A gentleman distributing money among some poor
people, found he wanted lOs. to be able to give 5s. to each ;
therefore he gives 48, only, and finds that he has 5s. left : re-
quired the number of shillings and of poor people.
«flfM. 15 poor people, and 65 shillings.
127. There are two numbers whose sum is the 6th part of
their product, and the greater is to the less as 3 to 2. Required
those numbers. «^fis. 15 and 10.
JV*. B. This question may be solved likewise by means of one
unknown letter.
128. To find three numbers, such that the first, with half the
other two, the second with one third of the other two, and the
third with one fourth of the other two, may be equal to 34.
Ms. 26, 22, and 10.
129. To find a number consisting of three places, whose
digits are in arithmetical progression ; if this number be divided
by the sum of its digits, the quotient will he 48 ; and if from
the number be subtracted 198, the digits will be inverted.
Ms, 432.
130. To find three numbers such, that | the first, ^ of the se-
cond, and I of the third, shall be equal to 62 ; | of the first, | of
the second, and ^ of the third, equal fo 47 ; and \ of the firsts
^ of the second, and | of the third, equal to 38.
Jhis. 24, 60, 120.
131. To find three numbera such that the first with | of the
\
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^uesKoM for Practice. fl
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sum of the second and third shall be 1£0, the second w^^ \ c
the difference of the third and first shall be 70, and | of the suri
of the three numbers shall be 95. JSm. 50, 65^ 75.
132. What is that fraction which will become equal to ^, i
an unit be added to the numerator ; but on the contrary* if a
unit be added to the denominator^ it will be equal to ^ ?
dm. -^j.
133. The dimensions of a certain rectangular floor are suet
that if it had been 2 feet broader^ and 3 feet longer* it woul
have been 64 square feet larger $ but if it had been 3 feet broad
er and 2 feet longer* it would then have been 68 square feet lar^
er : required the length and breadth of the floor.
Jn$, Length 14 feet* and breadth 10 feet.
134. A person found that upon beginning the study of hi
profession ^ of his life hitherto had passed before he commence
his education* 4 under a private teacher* and the same time at
public school* and four years at the , university. What was hi
age ? Jins. 21 years.
135. To find a number such that whether it be divided int
two or three equal parts the continued product of the parts sha
be equal to the same quantity. Jhts. 6|.
136. There is a certain number* consisting of two digits
The sum of these digits is 5* and if 9 be added to the numbe
itself the digits will be inverted. What is the number ?
An*. 23.
13T. What number is that* to which if I add 20 and from | (
this sum I subtract 12* the remainder shall be 10 ? Ms. 13.
^uadralic Equations.
SECTIOir IV. CHAPTER 5.
138. To find that number to which 20 being added* and froi
which 10 being subtracted* the square of the sum* added t
twice the square of the remainder* shall be 17475. dns. 75.
139. What two numbers are those* which are to one anothc
in the ratio of 3 to 5, and whose squares^ added iogether* mak
1666 7 .ins. 21 and 35.
n
216
JUgebra.
140. The sum S a, and the sam of the squares 2 bf of two niuii-
bers being given j to find the numbers.
Ms. a — s/b a* anda-fvft ^t .
141. To divide the number 100 into two such parts, that the
sum of their square roots may be 14. Jins. 64 and 36.
142. To find three such numbers^ that the sum of the first
and second multiplied into the third, may be equal to 6S ; aad
the sum of the second and third, multiplied into the first equd
to 28 ; also, that the sum of the first and third, multiplied into
the second, may be equal to 55. JBhis. 2, 5, 9.
143. What two numbers are thdse, whose sum is to tbo
greater as 11 to 7 j the difference of their squares being 158?
Jim. 14 and 8.
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